On the Commuting Graph of Semidihedral Group

Abstract

The commuting graph $\varDelta (G)$ of a finite non-abelian group G is a simple graph with vertex set G, and two distinct vertices x, y are adjacent if $xy = yx$. In this paper, first we discuss some properties of $\varDelta (G)$. We determine the edge connectivity and the minimum degree of $\varDelta (G)$ and prove that both are equal. Then, other graph invariants, namely: matching number, clique number, boundary vertex, of $\varDelta (G)$ are studied. Also, we give necessary and sufficient condition on the group G such that the interior and center of $\varDelta (G)$ are equal. Further, we investigate the commuting graph of the semidihedral group $SD_{8n}$. In this connection, we discuss various graph invariants of $\varDelta (SD_{8n})$ including vertex connectivity, independence number, matching number and detour properties. We also obtain the Laplacian spectrum, metric dimension and resolving polynomial of $\varDelta (SD_{8n})$.

1 Introduction

The study of various algebraic structures through their graph-theoretic properties becomes an interesting research topic. The investigation of graphs related to various algebraic constructions is very important because graphs of this type have valuable applications [13, 36] and are related to automata theory [26, 27]. Let G be a non-abelian group and $\varOmega \subseteq G$. The commuting graph of G, denoted by $\varDelta (G, \varOmega )$, is a simple graph in which the vertex set is $\varOmega$ and two distinct vertices $x, y \in \varOmega$ are adjacent whenever $xy = yx$. The commuting graphs were introduced by Brauer and Fowler [7] with $\varOmega = G {\setminus } \{e\}$. For $G = \varOmega$, we denote $\varDelta (G, G)$ by $\varDelta (G)$. Many authors have been studied commuting graphs for different choices of G and $\varOmega$ (see [9, 19, 31]). Moreover, [37,38,39] use combinatorial parameters of certain commuting graphs to establish long standing conjectures in the theory of division algebras. The investigation of graph invariants, namely: spectrum, metric dimension, detour distance, etc., becomes interesting and important because of their valuable applications. The Laplacian spectrum has many applications in chemistry, whereas the spectral radius is used in computer networks to protect personal data in some database. The authors of [14, 35] studied the Laplacian spectrum of certain graphs on algebraic structures. Slatter [41] introduced the concept of metric dimension and then separately studied by Harary et al. in [21]. The metric dimension of a graph is widely applicable in pharmaceutical chemistry [10, 13], robot navigation [28], etc. The metric dimension of some graphs associated with algebraic structures has been studied in [18, 20, 30]. The concept of detour distance was introduced by Chartrand et al. [11] and has applications in channel assignment in FM. In particular, detour index is widely used in chemical theory, see [45] and references therein. The concept of strong metric dimension has been introduced by Sebő et al. [36]. Strong metric dimension of graphs and their complements has been investigated in [44]. The strong metric dimension of graphs associated with algebraic structures has been studied in [29, 32, 33].

In recent years, the commuting graphs of various algebraic structures have become a topic of research for many mathematicians, see [2, 3, 17, 40] and references therein. Araújo et al. [4] calculated the diameter of commuting graphs of various ideals of full transformation semigroup. Also, for every natural number $n \ge 2$, a finite semigroup whose commuting graph has diameter n has been constructed in [4]. Iranmanesh et al. [24] studied diameter, girth, clique number, independence number, etc., of the commuting graph associated with symmetric group and alternating group. Let T be a transversal of the center Z(G) of a finite non-abelian group G. For $\varOmega = T {\setminus } Z(G)$, Julio et al. [34] proved that the commuting graph $\varDelta (G, \varOmega )$ is connected strongly regular if and only if G is isoclinic to an extraspecial 2-group of order at least 32. Further, they classified the groups G such that $\varDelta (G, \varOmega )$ is disconnected strongly regular. Tolue [42] introduced the twin non-commuting graph by partitioning the vertices of non-commuting graph and studied the graph theoretic properties of twin non-commuting graph of AC-group and dihedral group. The distant properties as well as detour distant properties of the commuting graph on the dihedral group $D_{2n}$ were investigated by Faisal et al. [2]. Moreover, they obtained metric dimension of the commuting graph on $D_{2n}$ and its resolving polynomial. Ali et al. [1] studied the connectivity and the spectral radius of the commuting graph of dihedral and dicyclic groups. Recently, Vipul et al. [25] studied the detour distance properties and obtained the resolving polynomial of the commuting graph of generalized dihedral group. Motivated with the work of [1, 2, 25], in this paper we consider the commuting graphs in the context of semidihedral group $SD_{8n}$.

This paper is structured as follows. In Sect. 2, we provide necessary background material and fix our notations used throughout the paper. In Sect. 3, we study some properties of $\varDelta (G)$. Section 4 comprises the study of various graph invariants of $\varDelta (SD_{8n})$, viz. Hamiltonian, perfectness, independence number, clique number, vertex connectivity, edge connectivity, vertex covering number, edge covering number, etc. Moreover, we study the Laplacian spectrum, metric dimension, resolving polynomial and the detour properties of the commuting graph of $SD_{8n}$.

2 Preliminaries

In this section, we recall necessary definitions, results and notations of graph theory from [43]. A graph $\varGamma$ is a pair $\varGamma = (V, E)$, where $V = V(\varGamma )$ and $E = E(\varGamma )$ are the set of vertices and edges of $\varGamma$, respectively. Moreover, the order of a graph $\varGamma$ is the number of vertices in $\varGamma$. We say that two different vertices a, b are $adjacent$, denoted by $a \sim b$, if there is an edge between a and b. Also, we denote this edge $a \sim b$ by (a, b). The neighborhood $\mathrm{N}(x)$ of a vertex x is the set of all vertices adjacent to x in $\varGamma$. Additionally, we denote $\mathrm{N}[x] = \mathrm{N}(x) \cup \{x\}$. It is clear that we are considering simple graphs, i.e., undirected graphs with no loops or repeated edges. If a and b are not adjacent, then we write $a \not \sim b$. A subgraph of a graph $\varGamma$ is a graph $\varGamma '$ such that $V(\varGamma ') \subseteq V(\varGamma )$ and $E(\varGamma ') \subseteq E(\varGamma )$. A walk $\lambda$ in $\varGamma$ from the vertex u to the vertex w is a sequence of vertices $u = v_1, v_2,\ldots , v_{m} = w$ $(m > 1)$ such that $v_i \sim v_{i + 1}$ for every $i \in \{1, 2, \ldots , m-1\}$. If no edge is repeated in $\lambda$, then it is called a trail in $\varGamma$. A trail whose initial and end vertices are identical is called a closed trail. A walk is said to be a path if no vertex is repeated. The length of a path is the number of edges it contains. A graph $\varGamma$ is said to be connected if there is a path between every pair of vertices. A graph $\varGamma$ is said to be complete if any two distinct vertices are adjacent. If $U \subseteq V(\varGamma )$, then the subgraph of $\varGamma$ induced by U is the graph $\varGamma '$ with vertex set U, and with two vertices adjacent in $\varGamma '$ if and only if they are adjacent in $\varGamma$. Moreover, $K_U$ denotes a complete subgraph induced by U. A path that begins and ends on the same vertex is called a cycle. A cycle C in a graph $\varGamma$ that includes every vertex of $\varGamma$ is called a Hamiltonian cycle of $\varGamma$. If $\varGamma$ contains a Hamiltonian cycle, then $\varGamma$ is called a Hamiltonian graph. The degree of a vertex v is the number of edges incident to v, and it is denoted as deg(v). The smallest degree among the vertices of $\varGamma$ is called the minimum degree of $\varGamma$, and it is denoted by $\delta (\varGamma )$. The chromatic number $\chi (\varGamma )$ of a graph $\varGamma$ is the smallest positive integer k such that the vertices of $\varGamma$ can be colored in k colors so that no two adjacent vertices share the same color. A graph $\varGamma$ is Eulerian if $\varGamma$ is both connected and has a closed trail (walk with no repeated edge) containing all the edges of a graph. Suppose $\varGamma _1 = (V_1, E_1)$ and $\varGamma _2 = (V_2, E_2)$ are graphs with disjoint vertex sets. The union $\varGamma _1 \cup \varGamma _2$ is the graph with $V(\varGamma _1 \cup \varGamma _2) = V_1 \cup V_2$ and $E(\varGamma _1 \cup \varGamma _2) = E_1 \cup E_2$. For a positive integer n, we write $n \varGamma$ to denote the union of n disjoint copies of $\varGamma$. The join $\varGamma _1 \vee \varGamma _2$ is the graph with $V(\varGamma _1 \cup \varGamma _2) = V_1 \cup V_2$ and $E(\varGamma _1 \cup \varGamma _2) = E_1 \cup E_2 \cup \{(a, b) : a \in V_1, b \in V_2 \}$.

A clique of a graph $\varGamma$ is a complete subgraph of $\varGamma$ and the number of vertices in a clique of maximum size is called the clique number of $\varGamma$ and it is denoted by $\omega ({\varGamma })$. The graph $\varGamma$ is perfect if $\omega (\varGamma ') = \chi (\varGamma ')$ for every induced subgraph $\varGamma '$ of $\varGamma$. Recall that the complement ${\overline{\varGamma }}$ of $\varGamma$ is a graph with same vertex set as $\varGamma$ and distinct vertices u, v are adjacent in ${\overline{\varGamma }}$ if they are not adjacent in $\varGamma$. A subgraph $\varGamma '$ of $\varGamma$ is called hole if $\varGamma '$ is a cycle as an induced subgraph, and $\varGamma '$ is called an antihole of $\varGamma$ if $\overline{\varGamma '}$ is a hole in ${\overline{\varGamma }}$.

Theorem 1

[15] A finite graph $\varGamma$ is perfect if and only if it does not contain hole or antihole of odd length at least 5.

Remark 1

Let G be a finite group and x is a dominating vertex. Then, x does not belong to the vertex set of any hole of length greater than 3, or any antihole of $\varDelta (G)$.

An independent set of a graph $\varGamma$ is a subset of $V(\varGamma )$ such that no two vertices in the subset are adjacent in $\varGamma$. The independence number of $\varGamma$ is the maximum size of an independent set; it is denoted by $\alpha (\varGamma )$.

A vertex cut set in a connected graph $\varGamma$ is a set of vertices whose deletion increases the number of connected components of $\varGamma$. The vertex connectivity of a connected graph $\varGamma$ is the minimum size of a vertex cut set, and it is denoted by $\kappa (\varGamma )$ . For $k \ge 1$, graph $\varGamma$ is k-connected if $\kappa (\varGamma ) \ge k$. The edge cut set and edge connectivity can be defined analogously. It will be denoted as $\kappa '(\varGamma )$. It is well known that $\kappa (\varGamma ) \le \kappa '(\varGamma ) \le \delta (\varGamma )$. An edge cover of a graph $\varGamma$ is a set L of edges such that every vertex of $\varGamma$ is incident to some edge of L. The minimum cardinality of an edge cover in $\varGamma$ is called the edge covering number; it is denoted by $\beta '(\varGamma )$. A vertex cover of a graph $\varGamma$ is a set Q of vertices such that it contains at least one endpoint of every edge of $\varGamma$. The minimum cardinality of a vertex cover in $\varGamma$ is called the vertex covering number; it is denoted by $\beta (\varGamma )$. A matching in a graph $\varGamma$ is a set of edges with no share endpoints and the maximum cardinality of a matching is called the matching number and it is denoted by $\alpha '(\varGamma )$. We have the following equalities involving the above parameters.

Lemma 1

[43, p. 115] Consider a graph $\varGamma$.

• (i) $\alpha (\varGamma ) + \beta (\varGamma ) = |V(\varGamma )|$.

• (ii) If $\varGamma$ has no isolated vertices, then $\alpha '(\varGamma ) + \beta '(\varGamma ) = |V(\varGamma )|$.

The (detour) distance, $(d_D(u,v))$ d(u, v), between two vertices u and v in a graph $\varGamma$ is the length of (longest) shortest $u-v$ path in $\varGamma$. The (detour) eccentricity of a vertex u, denoted by $(ecc_D(u))$ ecc(u), is the maximum (detour) distance between u and any vertex of $\varGamma$. The minimum (detour) eccentricity among the vertices of $\varGamma$ is called the (detour) radius of $\varGamma$; it is denoted by $(rad_D(\varGamma ))$ $rad(\varGamma )$. The detour diameter of a graph $\varGamma$ is the maximum detour eccentricity in $\varGamma$, denoted by $diam_D(G)$. A vertex v is said to be eccentric vertex for u if $d(u, v) = ecc(u)$. A vertex v is said to be an eccentric vertex of the graph $\varGamma$ if v is an eccentric vertex for some vertex of $\varGamma$. A graph $\varGamma$ is said to be an eccentric graph if every vertex of $\varGamma$ is an eccentric vertex. The center of $\varGamma$ is a subgraph of $\varGamma$ induced by the vertices having minimum eccentricity, and it is denoted by $Cen(\varGamma )$. The closure of a graph $\varGamma$ of order n is the graph obtained from $\varGamma$ by recursively joining pairs of non-adjacent vertices whose sum of degree is at least n until no such pair remains and it is denoted by $Cl(\varGamma )$. The graph $\varGamma$ is said to be closed if $\varGamma = Cl(\varGamma )$ [12].

A vertex v in a graph $\varGamma$ is a boundary vertex of a vertex u if $d(u, w) \le d(u, v)$ for $w \in$ N(v), while a vertex v is a boundary vertex of a graph $\varGamma$ if v is a boundary vertex of some vertex of $\varGamma$. The subgraph $\varGamma$ induced by its boundary vertices is the boundary $\partial (\varGamma )$ of $\varGamma$. A vertex v is said to be a complete vertex if the subgraph induced by the neighbors of v is complete. A vertex v is said to be an interior vertex of a graph $\varGamma$ if for each $u \ne v$, there exists a vertex w and a path $u-w$ such that v lies in that path at the same distance from both u and w. A subgraph induced by the interior vertices of $\varGamma$ is called interior of $\varGamma$, and it is denoted by $Int(\varGamma )$.

Theorem 2

[12, p. 337] Let $\varGamma$ be a connected graph and $v \in V(\varGamma )$. Then, v is a complete vertex of $\varGamma$ if and only if v is a boundary vertex of x for all $x \in V(\varGamma ) {\setminus } \{v\}$.

Theorem 3

[12, p. 339] Let $\varGamma$ be a connected graph and $v \in V(\varGamma )$. Then, v is a boundary vertex of $\varGamma$ if and only if v is not an interior vertex of $\varGamma$.

For vertices u and v in a graph $\varGamma$, we say that z strongly resolves u and v if there exists a shortest path from z to u containing v, or a shortest path from z to v containing u. A subset U of $V(\varGamma )$ is a strong resolving set of $\varGamma$ if every pair of vertices of $\varGamma$ is strongly resolved by some vertex of U. The least cardinality of a strong resolving set of $\varGamma$ is called the strong metric dimension of $\varGamma$ and is denoted by ${\text {sdim}}(\varGamma )$. For vertices u and v in a graph $\varGamma$, we write $u\equiv v$ if $\mathrm{N}[u] = \mathrm{N}[v]$. Notice that $\equiv$ is an equivalence relation on $V(\varGamma )$. We denote by ${\widehat{v}}$ the $\equiv$-class containing a vertex v of $\varGamma$. Consider a graph ${\widehat{\varGamma }}$ whose vertex set is the set of all $\equiv$-classes, and vertices ${\widehat{u}}$ and ${\widehat{v}}$ are adjacent if u and v are adjacent in $\varGamma$. This graph is well defined because in $\varGamma$, $w \sim v$ for all $w \in {\widehat{u}}$ if and only if $u \sim v$. We observe that ${\widehat{\varGamma }}$ is isomorphic to the subgraph ${\mathcal {R}}_{\varGamma }$ of $\varGamma$ induced by a set of vertices consisting of exactly one element from each $\equiv$-class. Subsequently, we have the following result of [29] with $\omega ({\mathcal {R}}_{\varGamma })$ replaced by $\omega ({\widehat{\varGamma }})$.

Theorem 4

[29, Theorem 2.2] Let $\varGamma$ be a graph with diameter 2. Then, sdim$(\varGamma ) = |V(\varGamma )| - \omega ({\widehat{\varGamma }})$.

When $\varGamma = \varDelta (G)$ for some group G, we denote ${\widehat{\varGamma }}$ by ${\widehat{\varDelta }}(G)$. We refer the reader to [23] for group theoretic definitions and terminologies not mentioned in this paper.

3 Commuting Graph of a Finite Group

In this section, we investigate the commuting graph of an arbitrary group G. First, we show that the edge connectivity and the minimum degree of $\varDelta (G)$ are equal. For $a \in G$, let cl(a) be the conjugacy class of G containing a. The center of G is denoted by Z(G) and the centralizer of the element a is denoted by $C_G(a) = \{b \in G: ab = ba \}$. The following remark follows from the definition of $\varDelta (G)$.

Remark 2

In the commuting graph of a group G, we have N$[x] = C_G(x)$ for each $x \in G$.

Theorem 5

Let G be a finite group and $t =$ max$\{|cl(a)| : a \in G\}$. Then,

$$\begin{aligned}\kappa '(\varDelta (G)) = \delta (\varDelta (G)) = \frac{|G|}{t} - 1.\end{aligned}$$

Proof

In view of Remark 2, $\delta (\varDelta (G)) = r - 1$, where $r =$ min$\{|C_G(a)| : a \in G\}$. For a graph $\varGamma$, since $\kappa '(\varGamma ) \le \delta (\varGamma )$, we obtain $\kappa '(\varDelta (G)) \le r - 1$. By Menger’s theorem (cf. [6, Theorem 3.2]), to prove another inequality, it is sufficient to show that there exist at least $r - 1$ internally edge disjoint paths between arbitrary pair of vertices. Let x and y be the distinct pair of vertices in $\varDelta (G)$. Suppose $|C_G(x) \cap C_G(y)| = q$. For $z \in C_G(x) \cap C_G(y)$, we have $x \sim z$ and $y \sim z$. Then, $\varDelta (G)$ contains at least q internally edge disjoint paths between x and y. Further there exist $x_1, x_2, \ldots , x_{r-q -1} \in C_G(x) {\setminus } C_G(y)$ and $y_1, y_2, \ldots , y_{r-q - 1} \in C_G(y) {\setminus } C_G(x)$. Consequently, we get $x \sim x_i \sim e \sim y_i \sim y$ internally edge disjoint paths between x and y which are $r-q - 1$ in total. Thus, we have at least $r - 1$ internally disjoint paths between x and y. Since for $x \in G$, we have $|cl(x)| = \frac{|G|}{|C_G(x)|}$. Hence, $\kappa '(\varDelta (G)) = \delta (\varDelta (G)) = \frac{|G|}{t} - 1$, where $t =$ max$\{|cl(a)| : a \in G\}$. $\square$

In the following theorem, we obtain the matching number of $\varDelta (G)$.

Theorem 6

Let G be a finite group and let t be the number of involutions in $G {\setminus } Z(G)$. Then,

• (i) $\alpha '(\varDelta (G)) = \left\{ \begin{array}{ll} \dfrac{|G|-1}{2}, &{} \quad \text {if }|G| \text { is odd};\\ \dfrac{|G|}{2}, &{} \quad \text {if }|G|\text { is even and }t \le |Z(G)|.\end{array}\right.$

• (ii) for $t > |Z(G)|$ and G is of even order, we have

  $$\begin{aligned} \dfrac{|G| + |Z(G)| - t}{2} \le \alpha '(\varDelta (G)) \le \dfrac{|G|}{2}. \end{aligned}$$

Proof

• (i) Let G be a finite group of odd order. Observe that for $x\in G {{\setminus }} \{e\}$, we have $x \ne x^{-1}$ as $o(x) > 2$ and $x \sim x^{-1}$. Thus, $M = \{(x, x^{-1}) : x \ne e \in G\}$ is a matching of order $\dfrac{|G|-1}{2}$ in $\varDelta (G)$. On the other hand, the order of a largest matching in a graph of order n is $\left\lfloor \dfrac{n}{2} \right\rfloor$. Hence, we get $\alpha '(\varDelta (G)) = \dfrac{|G|-1}{2}$.

  Now we assume that G is of even order and $t \le |Z(G)|$. Note that $x \in Z(G)$ if and only if $x^{-1} \in Z(G)$. Consider the set $A = \{a \in G {\setminus } Z(G) : o(a) = 2\}$ whose cardinality is t. Further, we denote the edges with ends $a_i$ and $z_i$ by $\epsilon _i$, where $a_i \in A$ and $z_i \in Z(G)$. Let $M = \{\epsilon _i : 1 \le i \le t \} \bigcup \{(x, x^{-1}) : x \ne e \in G {\setminus } Z(G)\}$ is a matching such that $G {\setminus } G_M \subseteq Z(G)$, where $G_M = \{ x \in G : (\exists \; x' \in G), \; (x, x') \in M \}$. Clearly, $|G {\setminus } G_M|$ is even as both |G| and $|G_M|$ are even. Consequently, ${\mathcal {M}} = M \cup \{(x, x') : x \ne x' \; \text {and} \; x, x' \in G_M \}$ is a matching of size $\dfrac{|G|}{2}$. Since $\alpha '(\varDelta (G)) \le \dfrac{|G|}{2}$, we have the result.

• (ii) Suppose |G| is even and $t > |Z(G)|$. By the proof of part (i), we have a matching ${\mathcal {M}}$ of size at least $\dfrac{|G| + |Z(G)| - t}{2}$. Thus, we get the desired inequality. $\square$

In view of Lemma 1(ii), we have the following corollary.

Corollary 1

For a finite group G and let t be the number of involutions in $G {\setminus } Z(G)$, we have

• (i) $\beta '(\varDelta (G)) = \left\{ \begin{array}{ll} \dfrac{|G|+1}{2}, &{} \quad \text {if }|G| \text { is odd};\\ \\ \dfrac{|G|}{2}, &{} \quad \text {if }|G| \text { is even and }t \le |Z(G)|.\end{array}\right.$

• (ii) for $t > |Z(G)|$ and G is of even order, we have

  $$\begin{aligned} \dfrac{|G|}{2} \le \beta '(\varDelta (G)) \le \dfrac{|G| - |Z(G)| + t}{2} . \end{aligned}$$

For $x \in G {\setminus } Z(G)$, $C_G(x)$ is called maximal centralizer if there is no $y \in G {\setminus } Z(G)$ such that $C_G(x)$ is a proper subgroup of $C_G(y)$. In the following proposition, we compute the vertex connectivity of $\varDelta (G)$.

Proposition 1

Let G be a finite non-abelian group such that, for some $x \in G$, $C_G(x)$ is a maximal centralizer and an abelian subgroup of G. Then, $\kappa (\varDelta (G)) = |Z(G)|$.

Proof

Suppose $C_G(x)$ is an abelian subgroup of G for some $x \in G$. Clearly, $C_G(x) \ne G$ as G is a non-abelian group. For $y \in C_G(x)$, we have $zy = yz$ for all $z \in C_G(x)$ as $C_G(x)$ is an abelian subgroup of G gives $C_G(x) \subseteq C_G(y)$. Since $C_G(x)$ is a maximal centralizer so $C_G(x) = C_G(y)$ for all $y \in C_G(x) {\setminus } Z(G)$. As G is a non-abelian group, there exists $z \in G$ such that $xz \ne zx$. It follows that there is no path between x and z in the subgraph induced by the vertices of $G {\setminus } Z(G)$. For instance, if there is a path $x = x_1 \sim x_2 \sim \cdots \sim x_r = y$ for some $r > 1$, then it follows that $x_i \in C_G(x)$ for all i, where $1 \le i \le r$. Thus, $xz = zx$, a contradiction. Thus, the subgraph induced by the vertices of $G {\setminus } Z(G)$ is disconnected so $\kappa (\varDelta (G)) \le |Z(G)|$.

If there exists a vertex cut set ${\mathcal {O}}$ which do not contain Z(G), then there exists $a \in Z(G)$ such that $a \notin \mathcal O$. For distinct $x, y \in G {\setminus } {\mathcal {O}}$, we have $x \sim a \sim y$. Consequently, the subgroup induced by the vertices of $G {\setminus } {\mathcal {O}}$ is connected implies the set ${\mathcal {O}}$ is not a vertex cut set, a contradiction. Thus, any vertex cut set always contains Z(G). Consequently, $\kappa (\varDelta (G)) \ge |Z(G)|$ and hence $\kappa (\varDelta (G)) = |Z(G)|$. $\square$

A group G is called an AC-group if the centralizer of every non-central element is abelian.

Theorem 7

Let G be a finite group such that $\varDelta (G) \cong \varDelta (H)$ for some AC-group H. Then, G is an AC-group.

Proof

Suppose $\varDelta (G) \cong \varDelta (H)$, where H is an AC-group. By [19, Lemma 2.1], the subgraph induced by the vertices of $H {\setminus } Z(H)$ is $\bigcup \limits _{ i=1}^{r} K_{|X_i| - |Z(H)|}$, where $X_1, X_2, \ldots , X_r$ are the distinct centralizers of non-central elements of H. Note that for each $x \in Z(H)$, we get $x \sim y$ for all $y\in H$. Therefore, we have $\varDelta (G) \cong \varDelta (H) = K_{|Z(H)|} \vee \left( \bigcup \limits _{ i=1}^{r} K_{|X_i| - |Z(H)|}\right)$. If $x \in G {\setminus } Z(G)$, then clearly N$[x] = X_i$ for some i. The subgraph induced by the vertices of $X_i$ is complete follows that $C_G(x)$ is an abelian subgroup of G. Thus, G is an AC-group. $\square$

Proposition 2

Let K be a clique in $\varDelta (G)$. Then, $\omega (\varDelta (G)) = |K|$ if and only if K is a commutative subgroup of maximum size in G.

Proof

Let K be any clique of maximum size such that $x, y \in K$. Then, xy commutes to every element of K. Consequently, we get $xy \in K$ as |K| is maximum. Note that the identity element e of G is in K. If $x \in K$, then $x^{-1} \in \langle x \rangle$ so $x^{-1}$ commutes with every element of K. Since K is a clique of maximum size, we obtain $x^{-1} \in K$. Therefore, K forms a subgroup of G. Clearly, K is a commutative subgroup of maximum size. Converse part is straightforward. $\square$

The proof of the following lemma follows from the definition of complete vertex.

Lemma 2

An element x is a complete vertex in $\varDelta (G)$ if and only if $C_G(x)$ is a commutative subgroup of G.

Proposition 3

For any group G, we have

$$\begin{aligned} Ecc(\varDelta (G)) = \left\{ \begin{array}{ll} \varDelta (G) {\setminus } \{e\}, &{} \quad \hbox { if}\ |Z(G)| = 1;\\ \varDelta (G), &{} \quad \mathrm{otherwise.}\end{array}\right. \end{aligned}$$

Proof

Let $x \in G {\setminus } \{e\}$. Then, $d(x, e) = ecc(e)$ so that x is an eccentric vertex for e. Thus, each non-identity element of G is an eccentric vertex of $\varDelta (G)$. If $|Z(G)| > 1$, then there exists $x \in Z(G) {\setminus } \{e\}$. Note that e is an eccentric vertex for x. Thus, the result holds. For $Z(G) = \{e\}$, one can observe that e is not an eccentric vertex of $\varDelta (G)$. For instance, if e is an eccentric vertex for some $y \in G$, then $d(x, e) = 1 = ecc(y)$. As $y \in G {\setminus } Z(G)$, there exists $z \in G$ such that $z \not \sim y$ gives $ecc(y) > 1$, a contradiction. $\square$

Corollary 2

Let G be a group with $|Z(G)| > 1$. Then, $\varDelta (G)$ is an eccentric graph.

In the next lemma, for each $x \in G$, we obtain the condition on $y \in G$ such that y is a boundary vertex of x.

Lemma 3

An element y is a boundary vertex of x in $\varDelta (G)$ if and only if one of the following hold:

• (i) $y \notin C_G(x)$.

• (ii) $C_G(y) \subseteq C_G(x)$.

Proof

If $x \in Z(G)$, then by the definition of boundary vertex the result holds. Now, let $x \in G {\setminus } Z(G)$. Suppose y is a boundary vertex of x. On contrary, we assume that $y \in C_G(x)$ and $C_G(y) \nsubseteq C_G(x)$. Then, there exists $z \in C_G(y)$ such that $z \notin C_G(x)$. Consequently, we get $d(x, z) > 1$ and $d(x, y) = 1$, a contradiction. On the other hand, we assume that y satisfy either (i) or (ii). Suppose $y \notin C_G(x)$. Since e is adjacent to all the vertices of $\varDelta (G)$ so diameter of $\varDelta (G)$ is at most two. Therefore, $d(x, y) = 2$ as $x \not \sim y$ implies $d(x, y) \ge d(x, z)$ for all $z \in G$. Consequently, y is a boundary vertex of x. If $C_G(y) \subseteq C_G(x)$, then clearly y is a boundary vertex of x. $\square$

Proposition 4

For the graph $\varDelta (G)$, we have $\partial (\varDelta (G)) = Ecc(\varDelta (G)).$

Proof

For $x \ne e$, we have $C_G(x) \subseteq C_G(e)$, so by Lemma 3, x is a boundary vertex of e. Therefore, x is a boundary vertex of $\varDelta (G)$. If $|Z(G)| > 1$, there exists $x \ne e \in Z(G)$ so e is a boundary vertex of $\varDelta (G)$. For $Z(G) = \{e\}$, note that e is not a boundary vertex of $\varDelta (G)$. For instance, if e is a boundary vertex of x for some $x \in G {\setminus } \{e\}$, then d$(x, y) \le \mathrm{d}(x, e) =1$ for all $y \in \mathrm{N}(e) = G {\setminus } \{e\}$. Consequently, we get $x \in Z(G)$ which is not possible. Thus, by Proposition 3, the result holds. $\square$

Lemma 4

[2, Lemma 1.2] For any group G, $Cen(\varDelta (G))$ is the subgraph induced by the vertices of Z(G).

Further, we characterize the group G such that $Int(\varDelta (G)) = Cen(\varDelta (G))$.

Theorem 8

Let G be a non-abelian group with $|G| > 2$. Then, $Int(\varDelta (G)) = Cen(\varDelta (G))$ if and only if G is an AC-group.

Proof

In view of Lemma 4, we show that $Int(\varDelta (G))$ is the subgraph induced by the vertices of Z(G) if and only if G is an AC-group. First we assume that G is an AC-group. We claim that v is an interior point if and only if $v \in Z(G)$. Let $v \in G {\setminus } Z(G)$. Since $C_G(v)$ is a commutative subgroup of G as G is an AC-group so by Theorems 2, 3 and Lemma 2, v is not an interior point of $\varDelta (G)$. On the other hand, we assume that $v \in Z(G)$. Then, clearly v is not a complete vertex as G is a non-abelian group. In view of Theorems 2 and 3, v is an interior point. Thus, $Int(\varDelta (G))$ is the subgraph induced by the vertices of Z(G).

Suppose $Int(\varDelta (G))$ is the subgraph induced by the vertices of Z(G). Let $x \in G {\setminus } Z(G)$. Then, x is not an interior point in $\varDelta (G)$ implies x is a complete vertex (cf. Theorems 2, 3 and Lemma 2). Consequently, $C_G(x)$ is an abelian subgroup of G. Thus, G is an AC-group. $\square$

Since the dihedral group $D_{2n}$ is an AC-group, we have the following corollary.

Corollary 3

[2, Theorem 2.11] For $n > 1$, the interior and the center of the commuting graph of the dihedral group $D_{2n}$ are equal.

4 Commuting Graph of the Semidihedral Group $SD_{8n}$

In this section, we obtain various graph invariants of $\varDelta (SD_{8n})$, viz. vertex connectivity, independence number, edge connectivity, matching number, clique number, etc. As a consequence, we obtain the vertex covering number and the edge covering number of $\varDelta (SD_{8n})$. Further, we also study the Laplacian spectrum, resolving polynomial and the detour properties of $\varDelta (SD_{8n})$ in various subsections. For $n \ge 2$, the semidihedral group $SD_{8n}$ is a group of order 8n with presentation

$$\begin{aligned} SD_{8n} = \langle a, b : a^{4n} = b^2 = e, ba = a^{2n -1}b \rangle . \end{aligned}$$

First note that

$$\begin{aligned} ba^i = \left\{ \begin{array}{ll} a^{4n -i}b, &{} \quad \text {if }i \text { is even;}\\ a^{2n - i}b,&{} \quad \text {if }i \text { is odd.}\end{array} \right. \end{aligned}$$

Thus, every element of $SD_{8n} {{\setminus }} \langle a \rangle$ is of the form $a^ib$ for some $0 \le i \le 4n-1$. We denote the subgroups $H_i = \langle a^{2i}b \rangle = \{e, a^{2i}b\}$ and $T_j = \langle a^{2j + 1}b \rangle = \{e, a^{2n}, a^{2j +1}b, a^{2n + 2j +1}b\}$. Then, we have

$$\begin{aligned} SD_{8n} = \langle a \rangle \cup \left( \bigcup \limits _{ i=0}^{2n-1} H_i \right) \cup \left( \bigcup \limits _{ j= 0}^{n-1} T_{j}\right) . \end{aligned}$$

Further, $Z(SD_{8n}) = \left\{ \begin{array}{ll} \{e, a^{2n}\}, &{} \quad \text {when }n \text { is even,}\\ \{e, a^n, a^{2n}, a^{3n}\},&{} \quad \text {otherwise.}\end{array} \right.$

By Remark 2, we have the following lemma.

Lemma 5

In $\varDelta (SD_{8n})$,

• (i) for even n, we have N$[x]= SD_{8n}$ if and only if $x \in \{e, a^{2n}\}$.

• (ii) for odd n, we have N$[x]= SD_{8n}$ if and only if $x \in \{e, a^n, a^{2n}, a^{3n}\}$.

By [43, Theorem 1.2.26], we have the following corollary.

Corollary 4

The commuting graph of $SD_{8n}$ is not Eulerian.

The following remarks will be useful in the sequel.

Remark 3

For even n and $1 \le i \le 4n$, we have $a^ib$ commutes with $a^jb$ if and only if $j = 2n + i$.

Remark 4

For even n and $1 \le i \le 4n$, we have $a^ib$ commutes with $a^j$ if and only if $j \in \{ 2n, 4n\}$.

Remark 5

For odd n and $1 \le i \le 4n$, we have $a^ib$ commutes with $a^j$ if and only if $j \in \{n, 2n, 3n, 4n\}$.

Remark 6

For odd n and $1 \le i \le 4n$, we have $a^ib$ commutes with $a^jb$ if and only if $j \in \{n + i, 2n + i, 3n +i\}$.

In view of Remarks 3–6, we obtain the neighborhood of each vertex of $\varDelta (SD_{8n})$.

Lemma 6

In $\varDelta (SD_{8n})$, for even n, we have

• (i) N$[x] = \{e, a^{2n}, a^ib, a^{2n + i}b\}$ if and only if $x \in \{a^i b, a^{2n +i}b\}$, where $1 \le i \le 4n$.

• (ii) N$[x] = \langle a \rangle$ if and only if $x \in \langle a \rangle {\setminus } \{e, a^{2n}\}$.

Lemma 7

In $\varDelta (SD_{8n})$, for odd n, we have

• (i) N$[x] = \{e, a^{n}, a^{2n}, a^{3n}, a^ib, a^{n + i}b, a^{2n + i}b, a^{3n + i}b\}$ if and only if $x \in \{a^ib, a^{n + i} b, a^{2n + i}b, a^{3n + i}b\}$, where $1 \le i \le 4n$.

• (ii) N$[x] = \langle a \rangle$ if and only if $x \in \langle a \rangle {\setminus } \{e, a^n, a^{2n}, a^{3n}\}$.

In view of Section 3 and Lemmas 6, 7, we have the following proposition.

Proposition 5

The commuting graph of $SD_{8n}$ satisfies the following properties:

• (i) sdim$(\varDelta (SD_{8n}))$ is $8n-2$.

• (ii) $\omega (\varDelta (SD_{8n})) = 4n$.

• (iii) $Cen(\varDelta (SD_{8n})) = Int(\varDelta (SD_{8n}))$.

• (iv) $\varDelta (SD_{8n})$ is an eccentric graph.

• (v) $\varDelta (SD_{8n})$ is a closed graph.

• (vi) $\kappa '(\varDelta (SD_{8n})) = \left\{ \begin{array}{ll} 3, &{} \quad \text {if }n \text { is even; }\\ 7, &{} \quad \text {if }n \text { is odd.}\end{array}\right.$

Proof

• (i) One can observe that the graph ${\widehat{\varDelta }}(SD_{8n})$ is a star graph. Thus, by Theorem 4, the result holds.

• (ii) For $1 \le i \le 4n$, note that the element $a^ib$ is commute with at most eight elements of $\varDelta (SD_{8n})$. Since the commutative subgroup generated by a is of size 4n, it follows that any commutative subgroup of $SD_{8n}$ of maximum size does not contain the elements of the form $a^ib$. Thus, $\langle a \rangle$ is a commutative subgroup of $SD_{8n}$ of maximum size 4n and hence the result holds (cf. Proposition 2).

• (iii) For any $x \in SD_{8n} {\setminus } Z(SD_{8n})$, we have N$[x] = \langle x \rangle$. By Remark 2, $C_G(x)$ is a commutative subgroup of $SD_{8n}$. Thus, by Theorem 8, $Int(\varDelta (SD_{8n})) = Cen(\varDelta (SD_{8n}))$.

• (iv) Since $|Z(\varDelta (SD_{8n}))| > 1$ (see Lemma 5) so that by Proposition 3, the result holds.

• (v) Note that for non-adjacent vertices x and y, we have $|\text {N}(x)|+|\text {N}(y)| < |V(\varDelta (SD_{8n}))|= 8n$ (cf. Lemmas 6 and 7). Consequently, deg$(x) + \text {deg}(y) < |V(\varDelta (SD_{8n}))|$ for all non-adjacent vertices x and y. Thus, by [2, Lemma 2.15 ], the result hold.

• (vi) For even n, by Lemma 6 note that $\delta (\varDelta (SD_{8n})) =$ min$\{ |C_G(x)| : x \in SD_{8n} \} - 1 = 3$ and for odd n, by Lemma 7, note that $\delta (\varDelta (SD_{8n})) =$ min$\{ |C_G(x)| : x \in SD_{8n} \} - 1 = 7$. Thus, by Theorem 5, we have the result. $\square$

As a consequence of Lemmas 6 and 7, we have the following proposition.

Proposition 6

For $n \ge 1$, we have

$$\begin{aligned} \varDelta (SD_{8n})\cong \left\{ \begin{array}{ll} K_2 \vee (K_{4n -2} \cup 2nK_2), &{} \quad \text {if }n \text { is even;}\\ K_4 \vee (K_{4n -4} \cup nK_4), &{} \quad \text {if }n \text { is odd.}\end{array}\right. \end{aligned}$$

Now, we obtain the automorphism group of the commuting graph of $SD_{8n}$. An automorphism of a graph $\varGamma$ is a permutation f on $V(\varGamma )$ with the property that, for any vertices u and v, we have $uf \sim vf$ if and only if $u \sim v$. The set $\mathrm{Aut}(\varGamma )$ of all graph automorphisms of a graph $\varGamma$ forms a group with respect to composition of mappings. The symmetric group of degree n is denoted by $S_n$.

Theorem 9

[5, Theorem 2.2] Suppose $\varGamma = n_1\varGamma _1 \cup n_2\varGamma _2\cup \cdots \cup n_t \varGamma _t$ with $\varGamma _i \ne \varGamma _ j$ for $i \ne j$. Then Aut$(\varGamma ) = \mathrm{Aut}(\varGamma _1) \wr S_{n_1} \times \mathrm{Aut}(\varGamma _2) \wr S_{n_2} \times \cdots \times \mathrm{Aut}(\varGamma _t) \wr S_{n_t}$.

Remark 7

If A is the set of all vertices adjacent to every vertex in a graph $\varGamma$ and $\varGamma - A$ is the subgraph of $\varGamma$ induced by the vertices of $V(\varGamma ) -A$, then Aut$(\varGamma )$ is isomorphic to $S_{|A|} \times \mathrm{Aut}(\varGamma - A)$.

By Proposition 6 and Theorem 9, we have the following theorem.

Theorem 10

For $n \in {\mathbb {N}}$, we have

$$\begin{aligned} \mathrm{Aut}(\varDelta (SD_{8n})) = \left\{ \begin{array}{ll} S_2 \times \left( (S_{4n-2} \wr S_1) \times (S_2 \wr S_{2n}) \right) , &{} \quad \text {if }n \text { is even;}\\ S_4 \times \left( (S_{4n-4} \wr S_1) \times (S_4 \wr S_{n}) \right) , &{} \quad \text {if }n \text { is odd.}\end{array}\right. \end{aligned}$$

Next, we obtain the vertex connectivity, independence number and the matching number of $\varDelta (SD_{8n})$.

Theorem 11

In the graph $\varDelta (SD_{8n})$,

• (i) the vertex connectivity of $\varDelta (SD_{8n})$ is given below:

  $$\begin{aligned}\kappa (\varDelta (SD_{8n})) = \left\{ \begin{array}{ll} 2, &{} \quad \text {if }n \text { is even };\\ 4, &{} \quad \text {if }n \text { is odd.}\end{array}\right. \end{aligned}$$

• (ii) the independence number of $\varDelta (SD_{8n})$ is given below:

  $$\begin{aligned}\alpha (\varDelta (SD_{8n})) = \left\{ \begin{array}{ll} 1, &{} \quad \hbox { if}\ n = 1;\\ 2n + 1, &{} \quad \text {if }n \text { is even };\\ n + 1, &{} \quad \text {otherwise.}\end{array}\right. \end{aligned}$$

• (iii) the matching number of $\varDelta (SD_{8n})$ is 4n.

Proof

• (i) In view of Remark 2 and Lemmas 6, 7, $C_G(ab)$ is an abelian subgroup of $SD_{8n}$ and maximal centralizer. By Proposition 1, $\kappa (\varDelta (SD_{8n})) = |Z(SD_{8n})|$. Thus, we have the result.

• (ii) Suppose n is even. Consider the set $I = \{a^ib : 1 \le i \le 2n \} \cup \{a\}$. In view of Lemmas 5 and 6, I is an independent in $\varDelta (SD_{8n})$ of size $2n +1$. If there exists another independent set $I'$ such that $|I'| > 2n + 1$, then there exist $x, y \in I'$ such that either $x, y \in \langle a \rangle$ or $x, y \in \{a^ib, a^{2n +i}b\}$ for some i, where $1 \le i \le 2n$ as $SD_{8n} = \langle a \rangle \cup (\bigcup \limits _{i = 1}^{2n} \{a^ib, a^{2n +i}b \})$. In both cases, we have $x \sim y$ (see Lemma 6), a contradiction of the fact that $I'$ is an independent in $\varDelta (SD_{8n})$. Thus, the result holds.

  On the other hand, we assume that n is odd. By using Lemma 7 and similar to even n case, we get an independent set $I = \{a^ib : 1 \le i \le n \} \cup \{a\}$ of the maximum size $n +1$.

• (iii) In view of Lemmas 6 and 7, $a^ib \sim a^{2n + i}b$ for all i, where $1 \le i \le 2n$. Consider the set ${\mathcal {M}} = \{(a^ib, a^{2n +i}b) \in E(\varDelta (SD_{8n})) : 1 \le i \le 2n \} \cup \{(a^i, a^{2n +i}) \in E(\varDelta (SD_{8n})): 1 \le i \le 2n \}$ which forms a matching of size 4n. Consequently, we get $\alpha '(\varDelta (SD_{8n})) \ge 4n$. It is well known that $\alpha '(\varDelta (SD_{8n})) \le \frac{|V(\varDelta (SD_{8n}))|}{2} = 4n$. Thus, $\alpha '(\varDelta (SD_{8n})) = 4n$. $\square$

In view of Lemma 1, we have the following corollary.

Corollary 5

For $n \ge 1$,

• (i) the vertex covering number of $\varDelta (SD_{8n})$ is given below:

  $$\begin{aligned}\beta (\varDelta (SD_{8n})) = \left\{ \begin{array}{ll} 7, &{} \quad \hbox { if}\ n = 1;\\ 6n - 1, &{} \quad \text {if }n \text { is even; }\\ 7n - 1, &{} \quad \text {otherwise.}\end{array}\right. \end{aligned}$$

• (ii) the edge covering number of $\varDelta (SD_{8n})$ is 4n.

Now, we investigate perfectness and Hamiltonian property of $\varDelta (SD_{8n})$.

Theorem 12

The commuting graph of $SD_{8n}$ is perfect.

Proof

In view of Theorem 1, it is enough to show that $\varDelta (SD_{8n})$ does not contain a hole or antihole of odd length at least five. Note that neither hole nor antihole can contain any element of $Z(\varDelta (SD_{8n}))$ (cf. Remark 1). First suppose that $\varDelta (SD_{8n})$ contains a hole C given by $x_1 \sim x_2\sim \dots \sim x_{2l + 1} \sim x_1$, where $l \ge 2$. Note that any hole can contain at most two elements of $\langle a \rangle$; otherwise, C contains a triangle which is not possible. In view of Lemmas 6 and 7, N$[x] = \langle a \rangle$ if and only if $x \in \langle a \rangle {\setminus } Z(\varDelta (SD_{8n}))$. It follows that $x_i \notin \langle a \rangle$ for all i, where $1 \le i \le 2l + 1$. Consequently, we get $a^ib \in C$ for some i. If n is even, then we must have $a^ib \sim a^{2n + i}b$ in $\varDelta (SD_{8n})$ as N$[a^ib] =$N$[a^{2n +i}b] = \{e, a^{2n}, a^ib, a^{2n +i}b\}$ (cf. Lemma 6). As a result, $a^{i}b$ is adjacent with only one element in C, a contradiction. In case of odd n, there exist $x, y \in C \; \cap$ N$(a^ib)$. Note that N$[a^ib] =$N$[a^{n + i}b] =$N$[a^{2n + i}b] =$N$[a^{3n + i}b] = \{e, a^{n}, a^{2n}, a^{3n}, a^ib, a^{n + i}b, a^{2n + i}b, a^{3n + i}b\}$ (cf. Lemma 7) implies $x, y \in \{a^{n + i}b, a^{2n + i}b, a^{3n + i}b\}$. Therefore, we have x, y and $a^ib$ forms a triangle in C, a contradiction. Thus, $\varDelta (SD_{8n})$ does not contain any hole of odd length at least five.

Now assume that $C'$ is an antihole of length at least 5 in $\varDelta (SD_{8n})$, that is, we have a hole $y_1 \sim y_2 \sim \dots \sim y_{2l + 1} \sim y_1$, where $l \ge 2$, in ${\overline{\varDelta (SD_{8n})}}$. Clearly, $y_i \notin Z(\varDelta (SD_{8n}))$ for all i, where $1 \le i \le 2l + 1$. Suppose $y_i \in \langle a \rangle$ for some i. Then, clearly $y_{i-1}, y_{i + 1} \in SD_{8n} {\setminus } \langle a \rangle$, otherwise $y_i \sim y_{i-1}$ and $y_i \sim y_{i +1}$ in $\varDelta (SD_{8n})$, a contradiction. Further note that for $1 \le j \le 2l +1$ and $j \notin \{i-1, i, i+1\}$, we have $y_j \in \langle a \rangle$. For instance, if $y_j \in SD_{8n} {\setminus } \left( \langle a \rangle \cup \{y_{i -1}, y_i, y_{i + 1}\}\right)$ for some j, then $y_j \sim y_i$ in ${\overline{\varDelta (SD_{8n})}}$ as $y_i \notin Z(\varDelta (SD_{8n}))$ (see Lemmas 6 and 7), a contradiction. Therefore, there exists $y_j \in \langle a \rangle$ gives $y_j \sim y_{i-1}$ and $y_j \sim y_{i+1}$ in ${\overline{\varDelta (SD_{8n})}}$. As a result, $\{y_j, y_{i-1}, y_i, y_{i+1}\}$ forms a cycle of length four in ${\overline{\varDelta (SD_{8n})}}$, a contradiction. Thus, $y_i \notin \langle a \rangle$ for all i.

If n is even, then $a^ib \sim a^jb$ for all $j \ne 2n + i$ in ${\overline{\varDelta (SD_{8n})}}$ (see Lemma 6) implies $C'$ is not an antihole, again a contradiction. Now we assume that n is odd. Let $y_1 = a^ib$ for some i. Then, we have $y_3, y_4 \in$ N$(a^ib) = \{e, a^{n}, a^{2n}, a^{3n}, a^{n + i}b, a^{2n + i}b, a^{3n + i}b\}$ gives $y_3 \not \sim y_4$ in $C'$ (see Lemma 7), a contradiction. Thus, $\varDelta (SD_{8n})$ does not contain any antihole of odd length at least five. $\square$

Theorem 13

The commuting graph of $SD_{8n}$ is Hamiltonian if and only if $n \in \{1, 3\}$.

Proof

First suppose that n is even. By Lemma 6, $\{e, a^{2n}\}$ is a vertex cut set in $\varDelta (SD_{8n})$ so that by deletion of these vertices, the connected components of the subgraph induced by the vertices $V(\varDelta (SD_{8n})){\setminus } \{e, a^{2n}\}$ are $\{a^ib, a^{2n + i}b\}$ and $\langle a \rangle {\setminus } \{e, a^{2n}\}$ where $1 \le i \le 2n$. Therefore, it is impossible to construct Hamiltonian cycle in $\varDelta (SD_{8n})$. Thus, $\varDelta (SD_{8n})$ is not Hamiltonian graph when n is even. Now we assume that n is odd. By Lemma 7, $\{e, a^n, a^{2n}, a^{3n}\}$ is a vertex cut set in $\varDelta (SD_{8n})$ so that by deletion of these vertices, the connected components of the subgraph induced by the vertices $V(\varDelta (SD_{8n})){\setminus } \{e, a^n, a^{2n}, a^{3n}\}$ are $\{a^ib, a^{n + i}b, a^{2n + i}b, a^{3n + i}b \}$ and $\langle a \rangle {\setminus } \{e, a^n, a^{2n}, a^{3n}\}$ where $1 \le i \le n$ which are $n + 1$ in total. It follows that for the construction of Hamiltonian cycle in $\varDelta (SD_{8n})$, we required at least $n + 1$ element from the vertex cut set. Thus, for $n > 3$, $\varDelta (SD_{8n})$ is not Hamiltonian graph. For $n = 1, 3$, in view of Lemma 7, we have deg$(x) \ge \frac{|V(\varDelta (SD_{8n}))|}{2}$ for all $x \in SD_{8n}$. Thus, by [43, Theorem 7.2.8], $\varDelta (SD_{8n})$ is Hamiltonian. $\square$

4.1 Laplacian Spectrum

In this subsection, we investigate the Laplacian spectrum of $\varDelta (SD_{8n})$. Consequently, we provide the number of spanning trees of $\varDelta (SD_{8n})$. For a finite simple undirected graph $\varGamma$ with vertex set $V(\varGamma ) = \{v_1, v_2, \ldots , v_n\}$, the adjacency matrix $A(\varGamma )$ is the $n\times n$ matrix with (i, j)th entry is 1 if $v_i$ and $v_j$ are adjacent and 0 otherwise. We denote the diagonal matrix $D(\varGamma ) = \mathrm{diag}(d_1, d_2, \ldots , d_n)$ where $d_i$ is the degree of the vertex $v_i$ of $\varGamma$, $i = 1, 2, \ldots , n$. The Laplacian matrix $L(\varGamma )$ of $\varGamma$ is the matrix $D(\varGamma ) - A(\varGamma )$. The matrix $L(\varGamma )$ is symmetric and positive semidefinite, so that its eigenvalues are real and nonnegative. Furthermore, the sum of each row (column) of $L(\varGamma )$ is zero. Recall that the characteristic polynomial of $L(\varGamma )$ is denoted by $\varPhi (L(\varGamma ), x)$. The eigenvalues of $L(\varGamma )$ are called the Laplacian eigenvalues of $\varGamma$ and it is denoted by $\lambda _1(\varGamma ) \ge \lambda _2(\varGamma ) \ge \cdots \ge \lambda _n(\varGamma ) = 0$. Now let $\lambda _{n_1}(\varGamma ) \ge \lambda _{n_2}(\varGamma ) \ge \cdots \ge \lambda _{n_r}(\varGamma ) = 0$ be the distinct eigenvalues of $\varGamma$ with multiplicities $m_1, m_2, \ldots , m_r$, respectively. The Laplacian spectrum of $\varGamma$, that is, the spectrum of $L(\varGamma )$ is represented as $\displaystyle \begin{pmatrix} \lambda _{n_1}(\varGamma ) &{} \lambda _{n_2}(\varGamma ) &{} \cdots &{} \lambda _{n_r}(\varGamma )\\ m_1 &{} m_2 &{} \cdots &{} m_r\\ \end{pmatrix}$. We denote the matrix $J_n$ as the square matrix of order n having all the entries as 1 and $I_n$ is the identity matrix of order n. In the following theorem, we obtain the characteristic polynomial of $L(\varDelta (SD_{8n}))$.

Theorem 14

For even n, the characteristic polynomial of the of $\varDelta (SD_{8n})$ is given by

$$\begin{aligned}\varPhi (L(\varDelta (SD_{8n})), x) =x(x -8n)^2 (x -4)^{2n}(x -2)^{2n}(x-4n)^{4n -3}.\end{aligned}$$

Proof

The Laplacian matrix $L(\varDelta (SD_{8n}))$ is the $8n \times 8n$ matrix given below, where the rows and columns are indexed in order by the vertices $e = a^{4n}, a^{2n}, a, a^2, \ldots , a^{2n - 1},a^{2n + 1}, a^{2n + 2}, \ldots , a^{4n - 1}$ and then $ab, a^2b, \ldots , a^{4n}b$.

$$\begin{aligned}L(\varDelta (SD_{8n})) = \displaystyle \begin{pmatrix} 8n-1 &{} -1 &{} -1 &{} -1&{} \cdots \cdots &{} -1 &{}-1 &{} \cdots \cdots &{}-1 \\ -1 &{} 8n - 1 &{} -1 &{} -1&{} \cdots \cdots &{} -1 &{}-1 &{} \cdots \cdots &{}-1 \\ -1 &{} -1&{} &{} &{} &{} &{} &{} &{} \\ -1 &{} -1 &{} &{} A &{} &{} &{} &{} {\mathcal {O}} &{} &{} \\ \; \; \vdots &{} \; \;\vdots &{} &{} &{} &{} &{} &{} &{} &{} \\ \; \; \vdots &{} \; \; \vdots &{} &{} &{} &{} &{} &{} &{} &{} \\ -1 &{} -1 &{} &{} &{} &{} &{} &{} &{} &{} \\ -1 &{} -1 &{} &{} &{} &{} &{} &{} &{} &{} \\ \; \; \vdots &{} \; \; \vdots &{} &{} {\mathcal {O}}'&{} &{} &{} &{} B &{} &{} \\ \; \; \vdots &{} \; \; \vdots &{} &{} &{} &{} &{} &{} &{} &{} \\ -1 &{} -1&{} &{} &{} &{} &{} &{} &{} &{} \\ \end{pmatrix}\end{aligned}$$

where $A = 4n I_{4n -2} - J_{4n -2}$, $B = \displaystyle \begin{vmatrix} 3I_{2n}&-I_{2n}\\ -I_{2n}&3I_{2n} \end{vmatrix}$, ${\mathcal {O}}$ is the zero matrix of size $(4n - 2) \times (4n)$ and ${\mathcal {O}}'$ is the transpose matrix of ${\mathcal {O}}$. Then the characteristic polynomial of $L(\varDelta (SD_{8n}))$ is

$$\begin{aligned}&\varPhi (L(\varDelta (SD_{8n})), x)\\&\quad = \displaystyle \begin{vmatrix} x - (8n-1)&1&1&1&\cdots \cdots&1&1&\cdots \cdots&1 \\ 1&x - ( 8n - 1)&1&1&\cdots \cdots&1&1&\cdots \cdots&1 \\ 1&1&&&&\\ 1&1&(xI_{4n -2} - A)&&{\mathcal {O}}&\\ \vdots&\vdots&&&&\\ \vdots&\vdots&&&&\\ 1&1&&&&\\ 1&1&&&&\\ \vdots&\vdots&{\mathcal {O}}'&&(xI_{4n}-B)&\\ \vdots&\vdots&&&&\\ 1&1&&&&\\ \end{vmatrix}.\end{aligned}$$

Apply row operation $R_1 \rightarrow (x -1)R_1 - R_2 - \cdots -R_{8n}$ and then expand by using first row, we get

$$\begin{aligned}&\varPhi (L(\varDelta (SD_{8n})), x)\\&\quad = \frac{x(x -8n)}{(x - 1)}\displaystyle \begin{vmatrix} x - (8n-1)&1&1&\cdots \cdots&1&1&\cdots \cdots&1\\ 1&&&&\\ \vdots&(xI_{4n -2} - A)&&{\mathcal {O}}&\\ \vdots&&&&\\ 1&&&&\\ 1&&&&\\ \vdots&{\mathcal {O}}'&&(xI_{4n}-B)&\\ \vdots&&&&\\ 1&&&&\\ \end{vmatrix}. \end{aligned}$$

Again, apply row operation $R_1 \rightarrow (x -2)R_1 - R_2 - R_3 -\cdots - R_{8n -1}$ and then expand by using first row, we get

$$\begin{aligned}\varPhi (L(\varDelta (SD_{8n})), x) = \frac{x(x -8n)^2}{(x-2)} \displaystyle \begin{vmatrix} xI_{4n -2} - A&{\mathcal {O}}\\ {\mathcal {O}}'&xI_{4n} - B \end{vmatrix}.\end{aligned}$$

By using Schur’s decomposition theorem [16], we have

$$\begin{aligned}\varPhi (L(\varDelta (SD_{8n})), x) = \frac{x(x -8n)^2}{(x-2)} |xI_{4n -2} - A|\cdot |xI_{4n}-B|.\end{aligned}$$

Clearly, $|xI_{4n} - B| = \displaystyle \begin{vmatrix} (x - 3)I_{2n}&I_{2n}\\ I_{2n}&(x -3)I_{2n} \end{vmatrix}$. Again by using Schur’s decomposition theorem, we obtain

$$\begin{aligned}|xI_{4n} - B| = |(x- 3)I_{2n}| |(x- 3)I_{2n} - \frac{1}{(x - 3)}I_{2n}| = (x -4)^{2n}(x -2)^{2n}.\end{aligned}$$

Now we obtain $|xI_{4n -2} - A| = |xI_{4n -2} -(4n I_{4n -2} - J_{4n -2})|$. It is easy to compute the characteristic polynomial of the matrix $J_{4n -2}$ is $x^{4n -3}(x-4n + 2)$. It is well known that if $f(x) = 0$ is any polynomial and $\lambda$ is an eigenvalue of the matrix P, then $f(\lambda )$ is an eigenvalue of the matrix f(P). Consequently, the eigenvalues of the matrix A are 4n and 2. Note that if x is an eigenvector of $J_n$ corresponding to the eigenvalue 0, then x is also an eigenvector of the matrix A corresponding to eigenvalue 4n. Since dimension of the null space of $J_{4n -2}$ is $4n - 3$ so that the multiplicity of the eigenvalue 4n in the characteristic polynomial of the matrix A is $4\hbox {n}-3$. Thus, $|x I_{4\hbox {n}-2} - \hbox {A}|= (x-4n)^{4n -3} (x-2)$ and hence the result holds. $\square$

Corollary 6

For even n, the Laplacian spectrum of $\varDelta (SD_{8n})$ is given by

$$\begin{aligned}\displaystyle \begin{pmatrix} 0 &{} \quad \;2 &{} \quad \;4 &{} \quad \;4n &{} \quad \;8n\\ 1 &{} \quad \;2n &{} \quad \; 2n &{} \quad \;4n -3 &{} \quad \;2\\ \end{pmatrix}.\end{aligned}$$

By [8, Corollary 4.2], we have the following corollary.

Corollary 7

For even n, the number of spanning trees of $\varDelta (SD_{8n})$ is $2^{14n - 3}n^{4n - 2}$.

Theorem 15

For odd n, the characteristic polynomial of the Laplacian matrix of $\varDelta (SD_{8n})$ is given by

$$\begin{aligned}\varPhi (L(\varDelta (SD_{8n})), x) =x(x -8n)^4 (x -4)^{n}(x -8)^{3n}(x-4n)^{4n -5}.\end{aligned}$$

Proof

The Laplacian matrix $L(\varDelta (SD_{8n}))$ is the $8n \times 8n$ matrix given below, where the rows and columns are indexed by the vertices $e = a^{4n}, a^{3n}, a^{2n}, a^{n}, a, a^2, \ldots , a^{n - 1},a^{n + 1}, a^{n + 2}, \ldots , a^{2n - 1}, a^{2n + 1}, a^{2n + 2}, \ldots , a^{3n - 1}$, $a^{3n + 1}, a^{3n + 2}, \ldots , a^{4n - 1}$ and then $ab, a^2b, \ldots , a^{4n}b$.

$$\begin{aligned}L(\varDelta (SD_{8n})) = \displaystyle \begin{pmatrix} 8n-1 &{} -1 &{} -1 &{} -1&{} \cdots \cdots &{} -1 &{}-1 &{} \cdots \cdots &{}-1 \\ -1 &{} 8n - 1 &{} -1 &{} -1&{} \cdots \cdots &{} -1 &{}-1 &{} \cdots \cdots &{}-1 \\ -1 &{} - 1 &{}8n -1 &{} -1&{} \cdots \cdots &{} -1 &{}-1 &{} \cdots \cdots &{}-1 \\ -1 &{} - 1 &{} -1 &{}8n -1&{} \cdots \cdots &{} -1 &{}-1 &{} \cdots \cdots &{}-1 \\ -1 &{} -1&{} -1 &{} -1 &{} &{} &{} &{} &{} \\ -1 &{} -1&{} -1 &{} -1 &{} &{} &{} &{} &{} \\ -1 &{} -1 &{} -1 &{} -1 &{} A &{} &{} &{} {\mathcal {O}} &{} &{} \\ \; \; \vdots &{} \; \;\vdots &{} \; \;\vdots &{} \; \;\vdots &{} &{} &{} &{} &{} &{} \\ -1 &{} -1 &{} -1 &{} -1 &{} &{} &{} &{} &{} &{} \\ \; \; \vdots &{} \; \; \vdots &{}\; \;\vdots &{}\; \;\vdots &{} {\mathcal {O}}'&{} &{} &{} B &{} &{} \\ -1 &{} -1&{} -1 &{} -1 &{} &{} &{} &{} &{} &{} \\ \end{pmatrix}\end{aligned}$$

where $A = 4nI_{(4n - 4)} - J_{(4n - 4)}$, $B = \displaystyle \begin{pmatrix} 7I_{n} &{} -I_{n} &{} -I_{n} &{} -I_{n}\\ -I_{n} &{} 7I_{n} &{} -I_{n} &{} -I_{n}\\ -I_{n} &{} -I_{n} &{} 7I_{n} &{} -I_{n}\\ -I_{n} &{} -I_{n} &{} -I_{n} &{} 7I_{n}\\ \end{pmatrix}$, ${\mathcal {O}}$ and ${\mathcal {O}}'$ are defined in Theorem 14. Then, the characteristic polynomial of $L(\varDelta (SD_{8n}))$ is

$$\begin{aligned}&\varPhi (L(\varDelta (SD_{8n})), x)\\&\quad = \displaystyle \begin{vmatrix} x-(8n-1)&1&1&1&\cdots&1&1&\cdots 1 \\ 1&x-(8n-1)&1&1&\cdots&1&1&\cdots 1 \\ 1&1&x-(8n-1)&1&\cdots&1&1&\cdots 1 \\ 1&1&1&x-(8n-1)&\cdots&1&1&\cdots 1 \\ 1&1&1&1&&&\\ 1&1&1&1&xI- A&&{\mathcal {O}}&\\ \vdots&\vdots&\vdots&\vdots&&&\\ \vdots&\vdots&\vdots&\vdots&&&\\ 1&1&1&1&&&\\ 1&1&1&1&&&\\ \vdots&\vdots&\vdots&\vdots&{\mathcal {O}}'&&xI - B&\\ \vdots&\vdots&\vdots&\vdots&&&\\ 1&1&1&1&&&\\ \end{vmatrix}\end{aligned}$$

Apply the following row operations consecutively

• $R_1 \rightarrow (x -1)R_1 - R_2 - \cdots - R_{8n}$

• $R_2 \rightarrow (x -2)R_2 - R_3 - \cdots - R_{8n}$

• $R_3 \rightarrow (x -3)R_3 - R_4 - \cdots - R_{8n}$

• $R_4 \rightarrow (x -4)R_4 - R_5 - \cdots - R_{8n}$

and then expand, we get

$$\begin{aligned}\varPhi (L(\varDelta (SD_{8n})), x) = \frac{x(x -8n)^4}{(x-4)} \displaystyle \begin{vmatrix} xI - A&{\mathcal {O}}\\ {\mathcal {O}}'&xI - B \end{vmatrix} = \frac{x(x -8n)^4}{(x-4)} |xI-A||xI-B|.\end{aligned}$$

By the similar argument used in the proof of Theorem 14, we obtain $|xI - A| = (x-4n)^{4n -5} (x-4)$. To get

$$\begin{aligned}|xI-B| = \displaystyle \begin{vmatrix} (x-7)I_{n}&I_{n}&I_{n}&I_{n}\\ I_{n}&(x-7)I_{n}&I_{n}&I_{n}\\ I_{n}&I_{n}&(x-7)I_{n}&I_{n}\\ I_{n}&I_{n}&I_{n}&(x-7)I_{n}\\ \end{vmatrix},\end{aligned}$$

apply the following row operations consecutively $R_i \rightarrow (x -5)R_i - R_{i +1} - \cdots - R_{4n}$ where $1 \le i \le n$ and then on solving, we get

$$\begin{aligned}|xI-B| = \frac{(x -4)^n (x-8)^n}{(x-5)^n} \displaystyle \begin{vmatrix} (x-7)I_{n}&I_{n}&I_{n}\\ I_{n}&(x-7)I_{n}&I_{n}\\ I_{n}&I_{n}&(x-7)I_{n}\\ \end{vmatrix}.\end{aligned}$$

Again apply the following row operations consecutively

• For $1 \le i \le n$, $R_i \rightarrow (x -6)R_i - R_{i +1} - \cdots - R_{3n}$

• For $n + 1 \le i \le 2n$ , $R_i \rightarrow (x -7)R_i - R_{i +1} - \cdots - R_{3n}$

and then expand, we obtain

$$\begin{aligned}|xI-B| = \frac{(x -4)^n (x-8)^{3n}}{(x-7)^n} |(x-7)I_n| = (x -4)^n (x-8)^{3n}.\end{aligned}$$

Thus, the result holds. $\square$

Corollary 8

For odd n, the Laplacian spectrum of $\varDelta (SD_{8n})$ is given by

$$\begin{aligned}\displaystyle \begin{pmatrix} 0 &{} \quad \; 4 &{} \quad \;8 &{} \quad \; 4n &{} \quad \; 8n\\ 1 &{} \quad \; n &{} \quad \; 3n &{} \quad \; 4n -5 &{} \quad \; 4\\ \end{pmatrix}.\end{aligned}$$

By [8, Corollary 4.2], we have the following corollary.

Corollary 9

For odd n, the number of spanning trees of $\varDelta (SD_{8n})$ is $2^{19n - 1} n^{4n - 2}$.

4.2 Resolving Polynomial

In this subsection, we obtain the resolving polynomial of $\varDelta (SD_{8n})$. First, we recall some of the basic definitions and necessary results. For z in $\varGamma$, we say that z resolves u and v if $d(z, u) \ne d(z, v)$. A subset U of $V(\varGamma )$ is a resolving set of $\varGamma$ if every pair of vertices of $\varGamma$ is resolved by some vertex of U. The least cardinality of a resolving set of $\varGamma$ is called the metric dimension of $\varGamma$ and is denoted by ${\text {dim}}(\varGamma )$. An i-subset of $V(\varGamma )$ is a subset of $V(\varGamma )$ of cardinality i. Let ${\mathcal {R}}(\varGamma , i)$ be the family of resolving sets which are i-subsets and $r_i = |{\mathcal {R}}(\varGamma , i)|$. Then, we define the resolving polynomial of a graph $\varGamma$ of order n, denoted by $\beta (\varGamma , x)$ as $\beta (\varGamma , x) = \mathop {\sum }_{i= dim(\varGamma )}^{n} r_ix^i$. The sequence $(r_{dim(\varGamma )}, r_{dim(\varGamma ) + 1}, \ldots , r_n)$ of coefficients of $\beta (\varGamma , x)$ is called the resolving sequence. Two distinct vertices u and v are said to be true twins if N$[u] =$N[v]. Two distinct vertices u and v are said to be false twins if N$(u) =$N(v). If u and v are true twins or false twins, then u and v are twins. For more details on twin vertices, we refer the reader to [22]. A set $U \subseteq V(\varGamma )$ is said to be a twin set in $\varGamma$ if u, v are twins for every pair of distinct pair of vertices $u, v \in U$. In order to obtain the resolving polynomial $\beta (\varDelta (SD_{8n}), x)$, the following results will be useful.

Remark 8

([2, Remark 3.3]) If U is twin set in a connected graph $\varGamma$ of order n with $|U| = l \ge 2$, then every resolving set for $\varGamma$ contains at least $l-1$ vertices of U.

Proposition 7

[2, Proposition 3.5] Let $\varGamma$ be a connected graph of order n. Then, the only resolving set of cardinality n is the set $V(\varGamma )$ and a resolving set of cardinality $n-1$ can be chosen n possible different ways.

Proposition 8

The metric dimension of $\varDelta (SD_{8n})$ is given below:

$$\begin{aligned}\mathrm{dim}(\varDelta (SD_{8n})) = \left\{ \begin{array}{ll} 6n-2, &{} \quad \text {if }n \text { is even; }\\ 7n - 2, &{} \quad \text {otherwise.}\end{array}\right. \end{aligned}$$

Proof

First we assume that n is even. In view of Lemma 6, we get twin sets $\langle a \rangle {\setminus } \{e, a^{2n}\}, \{e, a^{2n}\}$ and $\{a^ib, a^{2n + i}b\}$ where $1 \le i \le 2n$. By Remark 8, any resolving set in $\varDelta (SD_{8n})$ contains at least $6n-2$ vertices. Now we provide a resolving set of size $6n-2$. By Lemma 6, one can verify that the set $R_{\mathrm{even}} = \{a^ib : 1\le i \le 2n \} \cup \{a^i : i \ne 1, 2n \}$ is a resolving set of size $6n - 2$. Consequently, $\mathrm{dim}(\varDelta (SD_{8n})) = 6n - 2$. We may now suppose that n is odd. By Lemma 7, note that $\langle a \rangle {\setminus } \{e, a^n, a^{2n}, a^{3n}\}, \{e, a^n a^{2n}, a^{3n}\}$ and $\{a^ib, a^{n +i}b, a^{2n + i}b, a^{n +3i}b\}$, where $1 \le i \le n$, are twin sets in $\varDelta (SD_{8n})$. In view of Remark 8, any resolving set in $\varDelta (SD_{8n})$ contains at least $7n-2$ vertices. Further, it is routine to verify that the set $R_{\mathrm{odd}} = \{a^ib, a^{n + i}b, a^{2n + i}b : 1 \le i \le n\} \cup \{a^i : i \ne 1, 2n \}$ is a resolving set of size $7n-2$. Thus, $\mathrm{dim}(\varDelta (SD_{8n})) = 7n - 2$. $\square$

Theorem 16

For even n, the resolving polynomial of $\varDelta (SD_{8n})$ is given below:

$$\begin{aligned}\beta (\varDelta (SD_{8n}), x) = x^{8n} + 8n x^{8n-1} + 2^{2n + 2}(2n - 1) x^ {6n-2} + \mathop {\sum }_{i= 6n-1}^{8n - 2} r_ix^i,\end{aligned}$$

where $r_i = 2^{8n - i} \left\{ \left( {\begin{array}{c}2n + 1\\ 8n - i\end{array}}\right) + (2n - 1) \left( {\begin{array}{c}2n + 1\\ 8n - i -1\end{array}}\right) \right\}$ for $6n -1 \le i \le 8n -2$.

Proof

In view of Proposition 8, we have dim$(\varDelta (SD_{8n})) = 6n - 2$. It is sufficient to find the resolving sequence $(r_{6n - 2}, r_{6n -1}, \ldots , r_{8n -2}, r_{8n -1}, r_{8n})$. By the proof of Proposition 8, any resolving set R satisfies the following:

• $|R \cap (\langle a \rangle {\setminus } \{e, a^{2n}\})| \ge 4n - 3$;

• $|R \cap \{e, a^{2n}\}| \ge 1$;

• $|R \cap \{a^ib, a^{2n + i}b\}| \ge 1$ where $1 \le i \le 2n$.

For $|R| = i\ge 6n -2$, there exist $v_1, v_2, \ldots , v_{8n-i} \in SD_{8n} {\setminus } R$. Therefore, we have one of the following:

• (i) $v_j \in \langle a \rangle {\setminus } \{e, a^{2n}\}$ for some j and $v_1, v_2, \ldots , v_{j-1}, v_{j+1}, v_{j+2}, \ldots , v_{8n-i} \in \left( \displaystyle \bigcup \limits _{i = 1}^{2n} \{a^ib, a^{2n +i}b\}\right) \cup \{e, a^{2n}\}$.

• (ii) $v_1, v_2, \ldots , v_{8n-i} \in \left( \displaystyle \bigcup \limits _{i = 1}^{2n} \{a^ib, a^{2n +i}b\}\right) \cup \{e, a^{2n}\}$.

For $i = 6n -2$, (ii) does not hold so $v_j \in \langle a \rangle {\setminus } \{e, a^{2n}\}$ and

$v_1, v_2 \ldots , v_{j-1}, v_{j+1}, v_{j+2}, \ldots , v_{8n-i} \in \left( \displaystyle \bigcup \limits _{i = 1}^{2n} \{a^ib, a^{2n +i}b\}\right) \cup \{e, a^{2n}\}$. Therefore, we obtain $r_{6n-2} = 2^{2n + 1}(4n - 2)$. Now for fixed $i, \; 6n - 1 \le i \le 8n - 2$, we get $r_i = 2^{8n - i} \left\{ \left( {\begin{array}{c}2n + 1\\ 8n - i\end{array}}\right) + (2n - 1) \left( {\begin{array}{c}2n + 1\\ 8n - i -1\end{array}}\right) \right\}$. By Proposition 7, $r_{8n-1}= 8n$ and $r_{8n} = 1$. $\square$

Theorem 17

For odd n, the resolving polynomial of $\varDelta (SD_{8n})$ is given below:

$$\begin{aligned}\beta (\varDelta (SD_{8n}), x) = x^{8n} + 8n x^{8n - 1} + 2^{2n + 4}(n - 1)x^{7n-2} +\mathop {\sum }_{i= 7n-1}^{8n - 2} r_ix^i,\end{aligned}$$

where $r_i = 2^{16n - 2i} \left\{ \left( {\begin{array}{c}n + 1\\ 8n - i\end{array}}\right) + (n - 1) \left( {\begin{array}{c}n + 1\\ 8n - i -1\end{array}}\right) \right\}$ for $7n -1 \le i \le 8n -2$.

Proof

In view of Proposition 8, we have dim$(\varDelta (SD_{8n})) = 7n - 2$. It is sufficient to find the resolving sequence $(r_{7n - 2}, r_{7n -1}, \ldots , r_{8n -2}, r_{8n -1}, r_{8n})$. By the proof of Proposition 8, any resolving set R satisfies the following:

• $|R \cap (\langle a \rangle {\setminus } \{e, a^n, a^{2n}, a^{3n}\})| \ge 4n - 5$;

• $|R \cap \{e, a^n, a^{2n}, a^{3n}\}| \ge 3$;

• $|R \cap \{a^ib, a^{n + i}b, a^{2n + i}b, a^{3n + i}b\}| \ge 3$ where $1 \le i \le n$.

For $|R| = i\ge 7n -2$, there exist $v_1, v_2, \ldots , v_{8n-i} \in SD_{8n} {\setminus } R$. Therefore, we have one of the following

• (i) $v_j \in \langle a \rangle {\setminus } \{e, a^n, a^{2n}, a^{3n}\}$ for some j and $v_1, v_2, \ldots , v_{j-1}, v_{j+1}, v_{j+2}, \ldots , v_{8n-i} \in \left( \displaystyle \bigcup \limits _{i = 1}^{n} \{a^ib, a^{n +i}b, a^{2n +i}b, a^{3n + i}b\}\right) \cup \{e, a^n, a^{2n}, a^{3n}\}$.

• (ii) $v_1, v_2, \ldots , v_{8n-i} \in \left( \displaystyle \bigcup \limits _{i = 1}^{n} \{a^ib, a^{n +i}b, a^{2n +i}b, a^{3n + i}b\}\right) \cup \{e, a^n, a^{2n}, a^{3n}\}$.

For $i = 7n -2$, (ii) does not hold, so $v_j \in \langle a \rangle {\setminus } \{e, a^{2n}\}$ and

$v_1, v_2, \ldots , v_{j-1}, v_{j+1}, v_{j+2}, \ldots , v_{8n-i} \in \left( \displaystyle \bigcup \limits _{i = 1}^{n} \{a^ib, a^{n +i}b, a^{2n +i}b, a^{3n + i}b\}\right) \cup \{e, a^n, a^{2n}, a^{3n}\}$. Therefore, we have $r_{7n-2} = 4^{n + 1}(4n - 4)$. Now for fixed $i, \; 7n - 1 \le i \le 8n - 2$, we get

$$\begin{aligned}r_i = 2^{16n - 2i} \left\{ \left( {\begin{array}{c}n + 1\\ 8n - i\end{array}}\right) + (n - 1) \left( {\begin{array}{c}n + 1\\ 8n - i -1\end{array}}\right) \right\} .\end{aligned}$$

By Proposition 7, $r_{8n-1}= 8n$ and $r_{8n} = 1$. $\square$

4.3 Detour Distance Properties

In this subsection, we study the detour distance properties of $\varDelta (SD_{8n})$ viz. detour radius, detour eccentricity, detour degree, detour degree sequence and detour distance degree sequence of each vertex.

Theorem 18

In $\varDelta (SD_{8n})$, we have for each $x \in Z(SD_{8n})$,

$$\begin{aligned}ecc_D(x) =\left\{ \begin{array}{ll} 4n + 1, &{} \quad \text {when }n \text { is even;}\\ 4n + 11, &{} \quad \text {when }n \text { is odd.}\end{array}\right. \end{aligned}$$

and for each $x \in SD_{8n} {\setminus } Z(SD_{8n})$,

$$\begin{aligned} ecc_D(x) =\left\{ \begin{array}{ll} 4n + 3, &{} \quad \text {when }n \text { is even;}\\ 4n + 15, &{} \quad \text {when }n \text { is odd.}\end{array}\right. \end{aligned}$$

Proof

We split our proof in two cases depend on n.

Case 1 n is even. First note that $x \sim y$ for $x \in Z(SD_{8n})$ and $y \in SD_{8n} {\setminus } \{x\}$; $x' \sim y'$ for all distinct $x', y' \in \langle a \rangle {\setminus } \{e, a^{2n}\}$; for each $1 \le i \le 4n$, $a^ib \sim a^{2n + i}b$, $a^ib \not \sim a^jb$ for all $j \ne 2n + i$, $a^ib \not \sim a^j$ for all $a^j \in \langle a \rangle {\setminus } \{e, a^{2n}\}$. Thus, we have (i) for each $x \in Z(SD_{8n})$, there is a x - y detour of length $4n -1$ for all $y \in Z(SD_{8n}) {\setminus } \{x\}$; a x - y detour of length $4n +1$ for all $y \in SD_{8n} {\setminus } Z(SD_{8n})$ as $Z(SD_{8n}) = \{e, a^{2n}\}$; (ii) for each $1 \le i \le 4n$, there is a $a^ib$ - $a^{2n+i}b$ detour of length $4n +1$; for distinct $1 \le i, j \le 4n$ and $j \ne 2n +i$, a $a^ib$ - $a^jb$ detour of length $4n +3$; for each $1 \le i \le 4n$ and for each $a^j \in \langle a \rangle {\setminus } Z(SD_{8n})$, a $a^ib$ - $a^j$ detour of length $4n +3$; and (iii) for distinct $1 \le i, j < 4n$ and $i, j \ne 2n$, there is a $a^i$ - $a^j$ detour of length $4n + 1$.

Case 2 n is odd. First note that $x \sim y$ for $x \in Z(SD_{8n})$ and $y \in SD_{8n} {\setminus } \{x\}$; $x' \sim y'$ for all distinct $x', y' \in \langle a \rangle {\setminus } \{e, a^n, a^{2n}, a^{3n}\}$; for each $1 \le i \le n$ and for each $j \in \{n +i, 2n +i, 3n +i\}$, $a^ib \sim a^{j}b$; for each $1 \le i \le 4n$, $a^ib \not \sim a^jb$ for all $j \notin \{n +i,2n +i, 3n +i\}$, $a^ib \not \sim a^j$ for all $a^j \in \langle a \rangle {\setminus } \{e, a^n, a^{2n}, a^{3n}\}$. Thus, we have (i) for each $x \in Z(SD_{8n})$, there is a x - y detour of length $4n +7$ for all $y \in Z(SD_{8n}) {\setminus } \{x\}$; a x - y detour of length $4n +11$ for all $y \in SD_{8n} {\setminus } Z(SD_{8n})$; (ii) for each $1 \le i \le 4n$, there is a $a^ib$ - $a^{j}b$ detour of length $4n +11$ for all $j \in \{n +i,2n +i, 3n +i\}$; for distinct $1 \le i \le 4n$ and $j \notin \{n +i,2n +i, 3n +i\}$, a $a^ib$ - $a^{j}b$ detour of length $4n +15$; for each $1 \le i \le 4n$ and for each $a^j \in \langle a \rangle {\setminus } Z(SD_{8n})$, a $a^ib$ - $a^j$ detour of length $4n +15$; and (iii) for distinct $1 \le i, j < 4n$ and $i, j \notin \{n, 2n, 3n, 4n\}$, there is a $a^i$ - $a^j$ detour of length $4n + 11$. $\square$

By the definition of $rad_D(\varDelta (SD_{8n}))$ and $diam_D(\varDelta (SD_{8n}))$, we have the following corollary.

Corollary 10

In $\varDelta (SD_{8n})$, we have

• (i) $rad_D(\varDelta (SD_{8n})) = \left\{ \begin{array}{ll} 4n + 1, &{} \quad \text {if }n \text { is even; }\\ 4n + 11, &{} \quad \text {if }n \text { is odd.}\end{array}\right.$

• (ii) $diam_D(\varDelta (SD_{8n})) = \left\{ \begin{array}{ll} 4n + 3, &{} \quad \text {if }n \text { is even };\\ 4n + 15, &{} \quad \text {if }n \text { is odd.}\end{array}\right.$

The detour degree $d_D(v)$ of v is the number |D(v)|, where $D(v) = \{u \in V(\varGamma ) : d_D(u, v) = ecc_D(v) \}$. The average detour degree $D_{av}(\varGamma )$ of a graph $\varGamma$ is the quotient of the sum of the detour degrees of all the vertices of $\varGamma$ and the order of $\varGamma$. The detour degrees of the vertices of a graph written in non-increasing order are said to be the detour degree sequence of graph $\varGamma$, denoted by $D(\varGamma )$. For a vertex $x \in V(\varGamma )$, we denote $D_i(x)$ be the number of vertices at a detour distance i from the vertex x, and then the sequence $D_0(x), D_1(x), D_2(x), \ldots , D_{ecc_D(x)}(x)$ is called detour distance degree sequence of a vertex x, denoted by $dds_D(x)$. In the remaining part of this paper, $(a^r, b^s, c^t)$ denote a occur r times, b occur s times and c occur t times in the sequence. Now we have the following remark.

Remark 9

([2, Remark 2.6]) In a graph $\varGamma$, we have

• (i) $D_0(v) = 1$ and $D_{ecc_D}(v) = d_D(v)$.

• (ii) The length of sequence $dds_D(v)$ is one more than the detour eccentricity of v.

• (iii) $\displaystyle \sum \limits _{i = 0}^{ecc_D(v)} D_i(v) = |\varGamma |$.

Proposition 9

In $\varDelta (SD_{8n})$, we have for each $x \in Z(SD_{8n})$

$$\begin{aligned}d_D(x) = \left\{ \begin{array}{ll} 8n - 2, &{} \quad \text {when }n \text { is even};\\ 8n -4, &{} \quad \text {when }n \text { is odd;}\end{array}\right. \end{aligned}$$

for each $1 \le i \le 4n$,

$$\begin{aligned}d_D(a^ib) = \left\{ \begin{array}{ll} 8n - 4, &{} \quad \text {when }n \text { is even};\\ 8n -8, &{} \quad \text {when }n \text { is odd;}\end{array}\right. \end{aligned}$$

and for each $x \in \langle a \rangle {\setminus } Z(SD_{8n})$, $d_D(x) = 4n$.

Proof

Let $x \in Z(SD_{8n})$. In view of Theorem 18, $ecc_D(x) = 4n +1$ when n is even. Otherwise $ecc_D(x) = 4n +11$. In each case, by the proof of Theorem 18, one can observe that $D(x) = SD_{8n} {\setminus } Z(SD_{8n})$. Similar to $x \in Z(SD_{8n})$, for $x \in \langle a \rangle {\setminus } Z(SD_{8n})$ we obtain $D(x) = SD_{8n} {\setminus } \langle a \rangle$ (cf. Theorem 18). Now let $x = a^ib$ for some i, where $1 \le i \le 4n$. Similar to $x \in Z(SD_{8n})$, when n is even, we get

$$\begin{aligned}D(a^ib) = \left( \{a^jb : 1\le j \le 4n\} {\setminus } \{a^ib, a^{2n +i}b\}\right) \cup \left( \langle a\rangle {\setminus } Z(SD_{8n}) \right) .\end{aligned}$$

and for odd n,

$$\begin{aligned}D(a^ib) = \left( \{a^jb : 1\le j \le 4n\} {\setminus } \{a^{n+i}b, a^{2n +i}b, a^{3n +i}b, a^{4n + i}b\}\right) \cup \left( \langle a\rangle {\setminus } Z(SD_{8n}) \right) .\end{aligned}$$

$\square$

Corollary 11

In $\varDelta (SD_{8n})$, we have

• (i)
  $$\begin{aligned}D(\varDelta (SD_{8n})) = \left\{ \begin{array}{ll} \left( (4n)^{4n -2},(8n - 4)^{4n}, (8n-2)^2\right) , &{} \quad \text {if }n \text { is even};\\ \\ \left( (4n)^{4n -4}, (8n-8)^{4n}, (8n-4)^4\right) , &{} \quad \text {if }n \text { is odd.} \end{array}\right. \end{aligned}$$
• (ii)
  $$\begin{aligned}D_{av}(\varDelta (SD_{8n})) = \left\{ \begin{array}{ll} \frac{12n^2 -2n -1}{2n} &{} \quad \text {if }n \text { is even};\\ \\ \frac{2(3n^2 -n -1)}{n} &{} \quad \text {if }n \text { is odd.} \end{array}\right. \end{aligned}$$

Theorem 19

In $\varDelta (SD_{8n})$, we have

$$\begin{aligned}&dds_D(\varDelta (SD_{8n}))\\&\quad = \left\{ \begin{array}{ll} (1,0^{4n-2}, 1, 0, 8n-2)^2, (1,0^{4n}, 4n -1, 0, 4n)^{4n-2}, (1, 0^{4n}, 3, 0, 8n - 4)^{4n},\\ \text {if }n \text { is even; }\\ \\ (1,0^{4n+6}, 3, 0^3, 8n-4)^4, (1,0^{4n +10}, 4n -1, 0^3, 4n)^{4n -4}, (1, 0^{4n +10}, 7, 0^3,8n - 8)^{4n}, \\ \text {if }n \text { is odd.}\end{array}\right. \end{aligned}$$

Proof

Case 1 n is even. For $x \in Z(SD_{8n})$, by the proof of Theorem 18 (Case 1), we have $ecc_D(x) = 4n +1$ so $dds_D(x) = (1,\underbrace{0, 0, \ldots , 0}_{4{n} -2},1, 0, 8n-2)$. For $x \in SD_{8n} {\setminus } Z(SD_{8n})$, again by the proof of Theorem 18 (Case 1), we have $ecc_D(x) = 4n +3$. Thus

$$\begin{aligned} dds_D(x) = \left\{ \begin{array}{ll} (1,\underbrace{0, 0, \ldots , 0}_{4\text {n}} 4n-1, 0, 4n), &{} \quad \hbox {if }x \in \langle a \rangle {\setminus } Z(SD_{8n});\\ (1,\underbrace{0, 0, \ldots , 0}_{4\text {n}} 3, 0, 8n-4), &{} \quad \text{ if } x \in SD_{8n} {\setminus } \langle a \rangle . \end{array}\right. \end{aligned}$$

Case 2 n is odd. For $x \in Z(SD_{8n})$, by the proof of Theorem 18 (Case1), we have $ecc_D(x) = 4n +11$ so $dds_D(x) = (1,\underbrace{0, 0, \ldots , 0}_{4\hbox {n} + 6}, 3, 0, 0, 0, 8n-4)$. For $x \in SD_{8n} {\setminus } Z(SD_{8n})$, again by the proof of Theorem 18 (Case 2), we have $ecc_D(x) = 4n + 15$. Consequently,

$$\begin{aligned} dds_D(x) = \left\{ \begin{array}{ll} (1,\underbrace{0, 0, \ldots , 0}_{4\text {n} + 10} ,4n - 1,0,0, 0, 4n), &{} \quad \hbox {if }x \in \langle a \rangle {\setminus } Z(SD_{8n});\\ (1,\underbrace{0, 0, \ldots , 0}_{4\text {n} + 10} ,7, 0, 0, 0, 8n-8), &{} \quad \hbox {if }x \in SD_{8n} {\setminus } \langle a \rangle . \end{array}\right. \end{aligned}$$

$\square$