The Total Eccentricity Sum of Non-adjacent Vertex Pairs in Graphs

Abstract

Classical topological indices, such as Zagreb indices ($M_{1}$ and $M_2$) and the well-studied eccentric connectivity index ($\xi ^{c}$) directly or indirectly consider the total contribution of all edges in a graph. By considering the total degree sum of all non-adjacent vertex pairs in a graph, Ashrafi et al. (Discrete Appl Math 158:1571–1578, 2010) proposed two new Zagreb-type indices, namely the first Zagreb coindex ($\overline{M}_{1}$) and second Zagreb coindex ($\overline{M}_{2}$), respectively. Motivated by Ashrafi et al., we consider the total eccentricity sum of all non-adjacent vertex pairs, which we call the eccentric connectivity coindex ($\overline{\xi }^{c}$), of a connected graph. In this paper, we study the extremal problems of $\overline{\xi }^{c}$ for connected graphs of given order, connected graphs of given order and size, and the trees, unicyclic graphs, bipartite graphs containing cycles and triangle-free graphs of given order, respectively. Additionally, we establish various lower bounds for $\overline{\xi }^{c}$ in terms of several other graph parameters.

1 Introduction

Throughout this paper, we consider only simple connected graphs. For a graph $G=(V,\,E)$ with vertex set $V=V(G)$ and edge set $E=E(G)$, the degree of a vertex v in G, denoted by $d_{G}(v)$, is the number of edges incident with v. Let $d_{G}(u,\,v)$ be the distance between vertices u and v in G. The eccentricity of a vertex v in a graph G is defined to be $\varepsilon _{G}(v)=\max \{d_{G}(u,\,v)|u\in V(G)\}$. The diameter of a connected graph is the greatest distance between any pair of vertices in this graph. A vertex subset S of a graph G is said to be an independent set of G, if the subgraph induced by S is an empty graph. An edge subset T of a graph G is said to be a matching of G, if any two edges in T do not share a common end vertex. Then, $\alpha =\max \{|S|: S \text{ is } \text{ an } \text{ independent } \text{ set } \text{ of } G\}$ and $\beta =\max \{|T|: T \text{ is } \text{ a } \text{ matching } \text{ of } G\}$ are said to be the independence number and matching number of G, respectively. The chromatic number of a graph G, denoted by $\chi (G)$, is the minimum number of colors needed to guarantee that G can be colored with these colors so that no two adjacent vertices have the same color. The vertex-connectivity$\kappa (G)$ (where G is not a complete graph) is the size of a minimal vertex-cut, and the edge-connectivity is $\kappa ^{'}(G)$ is the size of a smallest edge cut.

A topological index is a function defined on a (molecular) graph regardless of the labeling of its vertices. Till now, hundreds of different topological indices have been employed in QSAR/QSPR studies, some of which have been proved to be successful [25]. Among those successful topological indices, there are two degree-based topological indices, called the first Zagreb index and the second Zagreb index , which are defined to be

$$\begin{aligned} M_{1}(G)=\sum \limits _{u\in V(G)}\Big (d_{G}(u)\Big )^{2}\quad \text{ and } \quad M_{2}(G)=\sum \limits _{uv\in E(G)}d_{G}(u)d_{G}(v), \end{aligned}$$

respectively. During the past decades, a large amount of papers dealt with the properties of these two indices. For more details on Zagreb indices, see the recent papers [4, 5, 8,9,10,11,12, 15, 18, 21, 23, 24, 28, 29]. Recall that the first Zagreb index can be rewritten as

$$\begin{aligned} M_{1}(G)=\sum \limits _{uv\in E(G)}\Big (d_{G}(u)+d_{G}(v)\Big ). \end{aligned}$$

According to the above equality, Ashrafi et al. [1, 2] considered the total contribution of all non-adjacent vertex pairs in a graph, and they proposed two new Zagreb-type indices, namely the first Zagreb coindex and second Zagreb coindex , which are defined to be

$$\begin{aligned} \overline{M}_{1}(G)=\sum \limits _{uv\not \in E(G)}\Big (d_{G}(u)+d_{G}(v)\Big ) \quad \text{ and } \quad \overline{M}_{2}(G)= \sum \limits _{uv\not \in E(G)}d_{G}(u)d_{G}(v), \end{aligned}$$

respectively. For recent results on Zagreb coindices, see [1, 2, 16, 17].

The eccentric connectivity index of a connected graph G, denoted by $\xi ^{c}(G)$, is defined as

$$\begin{aligned} \xi ^{c}(G)=\sum \limits _{v\in V(G)}d_{G}(v)\varepsilon _{G}(v), \end{aligned}$$

where $\varepsilon _{G}(v)$ is the eccentricity of the vertex v.

The eccentric connectivity index is a graph invariant which can be used to predict biological and physical properties and has a vast potential in structure activity/property relationships see [13, 14, 19]. For the mathematical properties of this index, see [3, 20, 22] and the references cited therein. The eccentric connectivity index of a connected graph G can be rewritten as

$$\begin{aligned} \xi ^{c}(G)=\sum \limits _{uv\in E(G)}\Big (\varepsilon _{G}(u)+\varepsilon _{G}(v)\Big ). \end{aligned}$$

Motivated by Ashrafi et al.’s definition for Zagreb coindices, we consider the total eccentricity sum of all non-adjacent vertex pairs, which is defined for a connected graph G as

$$\begin{aligned} \overline{\xi }^{c}(G)=\sum \limits _{uv\not \in E(G)}\Big (\varepsilon _{G}(u)+\varepsilon _{G}(v)\Big ). \end{aligned}$$

(1)

Similar to Ashrafi et al.’s definition for Zagreb coindices, we call this new eccentricity-based graph invariant the eccentric connectivity coindex ($\overline{\xi }^{c}$).

By (1), we can rewrite $\overline{\xi }^{c}$ of a connected graph G as

$$\begin{aligned} \overline{\xi }^{c}(G)=\sum \limits _{u\in V(G)}\varepsilon _{G}(u)\Big ( n-1-d_{G}(u)\Big ). \end{aligned}$$

(2)

In this paper, we mainly study extremal properties of $\overline{\xi }^{c}$. We organize this paper as follows. In Sect. 2, we characterize all extremal graphs with the maximum and minimum $\overline{\xi }^c$, respectively, among all connected graphs of given order. In Sect. 3, we characterize the connected graph with given order, size and the minimum $\overline{\xi }^{c}$ as well as the tree, unicyclic graph, bipartite graph containing cycles and triangle-free graph with the minimum $\overline{\xi }^{c}$, respectively. In Sect. 4, we establish various lower bounds for $\overline{\xi }^{c}$ in terms of several other graph parameters including the number of pendent vertices, independence number, matching number, chromatic number, vertex-connectivity, and edge-connectivity.

Before proceeding, we introduce some further notation and terminology. A vertex in a graph is said to be a branch vertex if it is of degree no less than three. If the length of an internal path in a connected graph is equal to diameter, then it is said to be a diametrical path. Let G and H be two vertex-disjoint graphs. The join of graphs G and H, denoted by $G\vee H$, is defined as a graph whose vertex set is $V(G)\cup V(H)$ and edge set is $E(G)\cup E(H)\cup \{xy|x\in V(G), y\in V(H)\}$. Let $tK_{1}$ be the union of t copies of $K_{1}$. Denote by $T_{n,\,t}$ the Turán graph, a complete t-partite graph of order n with $|n_{i}-n_{j}|\le 1$, where $n_{i}$, $i=1,\,\ldots ,\,t$, is the number of vertices in the ith partite set of $T_{n,\,t}$. When $t=2$, $T_{n,\,2}$ is just the balanced bipartite graph $K_{\lfloor \frac{n}{2}\rfloor , \lceil \frac{n}{2}\rceil }$. Denoted by $P_{n}$, $S_{n}$ and $K_{n}$ the path, star and complete graph on n vertices, respectively. Let $K_{n}^{p}$ denote the graph obtained by attaching p pendent edges to a vertex of $K_{n-p}$. Other notation and terminology not defined here will conform to those in [7].

2 General Connected Graphs

In this section, we characterize all extremal graphs with the maximum and minimum $\overline{\xi }^c$, respectively, among all connected graphs of given order.

We first give two lemmas which will be used in the proof of our main results.

Lemma 2.1

Let G be a connected graph with at least three vertices.

(i):

If G is not isomorphic to $K_{n}$, then $\overline{\xi }^{c}(G)>\overline{\xi }^{c}(G+e)$, where $e\in E(\overline{G})$;

(ii):

If G has an edge e not being a cut edge, then $\overline{\xi }^{c}(G)<\overline{\xi }^{c}(G-e)$.

Proof

We first prove (i) holds. Suppose that G is not a complete graph. Then, there exists a pair of vertices u and v in G such that $uv\in E(\overline{G})$. It is obvious that $\varepsilon _{G}(x)\ge \varepsilon _{G+uv}(x)$ and $d_{G}(x)\le d_{G+uv}(x)$ for any vertex x in G. Also, we have $d_{G+uv}(u)> d_{G}(u)$. By (2), we have $\overline{\xi }^{c}(G)>\overline{\xi }^{c}(G+e)$, as claimed.

Now, we consider (ii). Suppose that e is not a cut edge in G. Since $G-e$ is connected and not the complete graph, by (i), we have $\overline{\xi }^{c}(G-e)> \overline{\xi }^{c}\Big ((G-e)+e\Big )=\overline{\xi }^{c}(G)$, as desired (Fig. 1). $\square$

Fig. 1

Operation I: $G_{1}\longrightarrow G_{2}$

Lemma 2.2

Suppose that $G_0$ is a nontrivial connected graph and u is a vertex in $G_0$. Let $G_{1}$ (resp., $G_{2}$) be a graph obtained by identifying the vertex u of $G_0$ with a non-pendent vertex $v_{i}$ (resp., a pendent vertex, say $v_{0}$,) of the path $P_{l+1}:v_{0}v_{1}\ldots ,\,v_{l}\,(l\ge 2)$, where $1\le i\le l-1$. If $\varepsilon _{G_{0}}(u)\ge \max \{l-i,\,i\}$, then $\overline{\xi }^{c}(G_{1})<\overline{\xi }^{c}(G_{2})$.

Proof

Suppose without loss of generality that $l-i\ge i$. Then, $\varepsilon _{G_{0}}(u)\ge l-i\ge i$. For each $x\in V(G_{0}){\setminus }\{u\}$, we have $\varepsilon _{G_{2}}(x)=\max \{\varepsilon _{G_{0}}(x),\, d_{G_{0}}(x,\,u)+l\}$, $\varepsilon _{G_{1}}(x)= \max \{\varepsilon _{G_{0}}(x),\, d_{G_{0}}(x,\,u)+l-i\}$ (as $l-i\ge i$). So, $\varepsilon _{G_{2}}(x)\ge \varepsilon _{G_{1}}(x)$ for each $x\in V(G_{0}){\setminus }\{u\}$. Also, $d_{G_{2}}(x)=d_{G_{1}}(x)$ for each $x\in V(G_{0}){\setminus }\{u\}$. Thus,

$$\begin{aligned} \sum \limits _{x\in V(G_{0}){\setminus } \{u\}}\varepsilon _{G_{2}}(x)\Big ( n-1-d_{G_2}(x)\Big )-\sum \limits _{x\in V(G_{0}){\setminus } \{u\}}\varepsilon _{G_{1}}(x)\Big ( n-1-d_{G_1}(x)\Big )\ge 0. \end{aligned}$$

(3)

We first assume that $i\ge 2$. Thus, $l\ge 2i\ge 4$.

For each $k=1,\,\ldots ,\,i-1$, we have $\varepsilon _{G_{2}}(v_{k})=\max \{k+\varepsilon _{G_{0}}(u),\, l-k\}\ge k+\varepsilon _{G_{0}}(u)$, $\varepsilon _{G_{1}}(v_{k})=\max \{i-k+\varepsilon _{G_{0}}(u),\, l-k\}=i-k+\varepsilon _{G_{0}}(u)$ (as $\varepsilon _{G_{0}}(u)\ge l-i$). Also, $d_{G_{2}}(v_{k})=d_{G_{1}}(v_{k})=2$.

Since $\varepsilon _{G_{2}}(v_{k})\ge k+\varepsilon _{G_{0}}(u)$, we have

$$\begin{aligned} \varepsilon _{G_{2}}(v_{k})-\varepsilon _{G_{1}}(v_{k})\ge \Big (k+\varepsilon _{G_{0}}(u)\Big )-\Big (i-k+\varepsilon _{G_{0}}(u)\Big )= 2k-i.\end{aligned}$$

So,

$$\begin{aligned}&\sum \limits _{k=1}^{i-1}\varepsilon _{G_{2}}(v_{k})\Big ( n-1-d_{G_{2}}(v_{k})\Big )-\sum \limits _{k=1}^{i-1}\varepsilon _{G_{1}}(v_{k})\Big ( n-1-d_{G_{1}}(v_{k})\Big )\nonumber \\= & {} (n-3)\sum \limits _{k=1}^{i-1}\Big (\varepsilon _{G_{2}}(v_{k})- \varepsilon _{G_{1}}(v_{k})\Big )\nonumber \\\ge & {} (n-3)\sum \limits _{k=1}^{\lfloor \frac{i}{2}\rfloor }(2k-i)\quad (\text{ as } 2k-i\ge 0 \text{ for } k\ge \frac{i}{2})\nonumber \\\ge & {} (n-3)\lfloor \frac{i}{2}\rfloor (2-i). \end{aligned}$$

(4)

For each $k=i+1,\,\ldots ,\,l-1$, we have $\varepsilon _{G_{2}}(v_{k})=\max \{k+\varepsilon _{G_{0}}(u),\, l-k\}$, $\varepsilon _{G_{1}}(v_{k})=\max \{k-i+\varepsilon _{G_{0}}(u),\, l-k,\,k\}$. Since $k+\varepsilon _{G_{0}}(u)\ge k+(l-i)\ge l+1>l-k$, then $\varepsilon _{G_{2}}(v_{k})=k+\varepsilon _{G_{0}}(u)$. Also, because $2k-i+\varepsilon _{G_{0}}(u)\ge k+1+\varepsilon _{G_{0}}(u)> i+\varepsilon _{G_{0}}(u)\ge l$, we get $k-i+\varepsilon _{G_{0}}(u)>l-k$. Moreover, $k-i+\varepsilon _{G_{0}}(u)\ge k-i+(l-i)\ge k$. Therefore, $\varepsilon _{G_{1}}(v_{k})=k-i+\varepsilon _{G_{0}}(u)$. Then,

$$\begin{aligned} \varepsilon _{G_{2}}(v_{k})=\varepsilon _{G_{1}}(v_{k})+i \end{aligned}$$

for each $k=i+1,\,\ldots ,\,l-1$. Also, for each $k=i+1,\,\ldots ,\,l-1$, $d_{G_{2}}(v_{k})=d_{G_{1}}(v_{k})=2$. So,

$$\begin{aligned}&\sum \limits _{k=i+1}^{l-1}\varepsilon _{G_{2}}(v_{k})\Big ( n-1-d_{G_{2}}(v_{k})\Big )-\sum \limits _{k=i+1}^{l-1}\varepsilon _{G_{1}}(v_{k})\Big ( n-1-d_{G_{1}}(v_{k})\Big )\nonumber \\= & {} (n-3)\sum \limits _{k=i+1}^{l-1}\Big (\varepsilon _{G_{2}}(v_{k})- \varepsilon _{G_{1}}(v_{k})\Big )\nonumber \\= & {} (n-3)(l-i-1)i. \end{aligned}$$

(5)

As $l\ge 2i$, we obtain $(l-i-1)i+\lfloor \frac{i}{2}\rfloor (2-i)\ge (l-i-1)i+\frac{i-1}{2}(2-i)\ge (i-1)i+\frac{i-1}{2}(2-i)=\frac{i(i+1)}{2}-1\ge 0$. This, in conjunction with (4) and (5), gives

$$\begin{aligned} \sum \limits _{k=1,\,k\ne i}^{l-1}\varepsilon _{G_{2}}(v_{k})\Big ( n-1-d_{G_{2}}(v_{k})\Big )-\sum \limits _{k=1,\,k\ne i}^{l-1}\varepsilon _{G_{1}}(v_{k})\Big ( n-1-d_{G_{1}}(v_{k})\Big )\ge 0.\nonumber \\ \end{aligned}$$

(6)

By (3) and (6), we have

$$\begin{aligned}&\sum \limits _{y\in V(G_{2}){\setminus } \{v_{0},\,v_{i},\,v_{l}\}}\varepsilon _{G_{2}}(y)\Big ( n-1-d_{G_{2}}(y)\Big )\nonumber \\&-\sum \limits _{y\in V(G_{1}){\setminus }\{v_{0},\,v_{i},\,v_{l}\}}\varepsilon _{G_{1}}(y)\Big ( n-1-d_{G_{1}}(y)\Big )\ge 0. \end{aligned}$$

(7)

Now, by (7), it suffices to prove that

$$\begin{aligned} \sum \limits _{y\in \{v_{0},\,v_{i},\,v_{l}\}}\varepsilon _{G_{2}}(y)\Big ( n-1-d_{G_{2}}(y)\Big )-\sum \limits _{y\in \{v_{0},\,v_{i},\,v_{l}\}}\varepsilon _{G_{1}}(y)\Big ( n-1-d_{G_{1}}(y)\Big )\ge 0. \end{aligned}$$

For vertex $v_{0}$, $\varepsilon _{G_{2}}(v_{0})=\max \{\varepsilon _{G_{0}}(u),\, l\}$, $\varepsilon _{G_{1}}(v_{0})=\max \{i+\varepsilon _{G_{0}}(u),\, l\}=i+\varepsilon _{G_{0}}(u)$ (as $\varepsilon _{G_{0}}(u)\ge l-i$), $d_{G_{2}}(v_{0})=1+d_{G_{0}}(u)$, $d_{G_{1}}(v_{0})=1$.

For vertex $v_{i}$, $\varepsilon _{G_{2}}(v_{i})=\max \{i+\varepsilon _{G_{0}}(u),\, l-i\}=i+\varepsilon _{G_{0}}(u)$, $\varepsilon _{G_{1}}(v_{i})=\max \{\varepsilon _{G_{0}}(u),\, l-i\}=\varepsilon _{G_{0}}(u)$, $d_{G_{2}}(v_{i})=2$, $d_{G_{1}}(v_{i})=d_{G_{0}}(u)+2$.

For vertex $v_{l}$, $\varepsilon _{G_{2}}(v_{l})=l+\varepsilon _{G_{0}}(u)$, $\varepsilon _{G_{1}}(v_{l})=\max \{l-i+\varepsilon _{G_{0}}(u),\, l\}=l-i+\varepsilon _{G_{0}}(u)$ (since $\varepsilon _{G_{0}}(u)\ge l-i \ge i$, we have $l-i+\varepsilon _{G_{0}}(u)\ge (l-i)+i= l$). Also, $d_{G_{2}}(v_{l})=d_{G_{1}}(v_{l})=1$.

By (2) and (7), we have

$$\begin{aligned} \overline{\xi }^{c}(G_{2})-\overline{\xi }^{c}(G_{1})\ge & {} \sum \limits _{y\in \{v_{0},\,v_{i},\,v_{l}\}}\varepsilon _{G_{2}}(y)( n-1-d_{G_{2}}(y))\nonumber \\&-\,\sum \limits _{y\in \{v_{0},\,v_{i},\,v_{l}\}}\varepsilon _{G_{1}}(y)( n-1-d_{G_{1}}(y))\nonumber \\= & {} \Big [(l+\varepsilon _{G_0}(u))( n-2)-(l-i+\varepsilon _{G_0}(u))( n-2)\Big ]\nonumber \\&+\,\Big [(i+\varepsilon _{G_0}(u))( n-3)-\varepsilon _{G_0}(u)( n-d_{G_0}(u)-3)\Big ]\nonumber \\&+\,\Big [\max \{l,\,\varepsilon _{G_0}(u)\}( n-d_{G_0}(u)-2)- (i+\varepsilon _{G_0}(u))( n-2)\Big ]\nonumber \\= & {} (n{-}2)i{+}(n{-}3)(i{+}\varepsilon _{G_0}(u)){-}(n{-}3)\varepsilon _{G_0}(u)+d_{G_0}(u)\varepsilon _{G_0}(u) \nonumber \\&+\,\max \{l,\,\varepsilon _{G_0}(u)\}( n-d_{G_0}(u)-2)- ( n-2)(i+\varepsilon _{G_0}(u))\nonumber \\= & {} (n-2)i-(i+\varepsilon _{G_0}(u))-(n-3)\varepsilon _{G_0}(u)+d_{G_0}(u)\varepsilon _{G_0}(u)\nonumber \\&+\,\max \{l,\,\varepsilon _{G_0}(u)\}( n-d_{G_0}(u)-2). \end{aligned}$$

(8)

We distinguish the following two cases.

Case 1$\varepsilon _{G_0}(u)\ge l$.

Then, by (8), we have

$$\begin{aligned} \overline{\xi }^{c}(G_{2})-\overline{\xi }^{c}(G_{1})\ge & {} (n-2)i-(i+\varepsilon _{G_0}(u))-(n-3)\varepsilon _{G_0}(u)+d_{G_0}(u)\varepsilon _{G_0}(u)\\&+\,\varepsilon _{G_0}(u)( n-d_{G_0}(u)-2)\\= & {} (n-3)i>0. \end{aligned}$$

Case 2$\varepsilon _{G_0}(u)<l$.

Then, by (8), we have

$$\begin{aligned} \overline{\xi }^{c}(G_{2})-\overline{\xi }^{c}(G_{1})\ge & {} (n-2)i-(i+\varepsilon _{G_0}(u))-(n-3)\varepsilon _{G_0}(u)+d_{G_0}(u)\varepsilon _{G_0}(u)\nonumber \\&+\,l( n-d_{G_0}(u)-2)\nonumber \\= & {} (n{-}3)i{+}(n-2)l{-}(n{-}2)\varepsilon _{G_0}(u)+d_{G_0}(u)\varepsilon _{G_0}(u)- ld_{G_0}(u)\nonumber \\> & {} (n-3)i+d_{G_0}(u)\varepsilon _{G_0}(u)- ld_{G_0}(u). \end{aligned}$$

(9)

By our assumption $\varepsilon _{G_{0}}(u)\ge l-i$, we have $i\ge l-\varepsilon _{G_{0}}(u)$. By (9), we have

$$\begin{aligned} \overline{\xi }^{c}(G_{2})-\overline{\xi }^{c}(G_{1})> & {} (n-3)( l-\varepsilon _{G_{0}}(u))+d_{G_0}(u)\varepsilon _{G_0}(u)- ld_{G_0}(u)\nonumber \\= & {} [d_{G_0}(u)-(n-3)](\varepsilon _{G_{0}}(u)-l). \end{aligned}$$

(10)

Clearly, $d_{G_0}(u)\le n-3$. By our assumption, $\varepsilon _{G_0}(u)<l$, we have $[d_{G_0}(u)-(n-3)](\varepsilon _{G_{0}}(u)-l)\ge 0$. So, $\overline{\xi }^{c}(G_{2})>\overline{\xi }^{c}(G_{1})$.

Summarizing above, when $i\ge 2$, $\overline{\xi }^{c}(G_{2})>\overline{\xi }^{c}(G_{1})$.

Now, we consider the case of $i=1$. If $l\ge 3$, by the same approach as above, we can prove that (5) holds. So, no matter whether $l=2$ or $l\ge 3$, by (3), we can prove that (7) holds. What remains to do is exactly the same as that used in the case of $i\ge 2$.

This completes the proof. $\square$

For graphs $G_{1}$ and $G_{2}$ as introduced in Lemma 2.2, we call the graph operation: $G_{1}\Longrightarrow G_{2}$ the Operation I on $G_{1}$.

By means of Lemmas 2.1 and 2.2, we are in a position to characterize connected graphs with the maximum and minimum $\overline{\xi }^{c}$, respectively. Our result is as follows.

Theorem 2.3

Among all connected graphs of order n, the graphs with the minimum and maximum $\overline{\xi }^{c}$ are $K_{n}$ and $P_{n}$, respectively.

Proof

The case of $n=2$ is trivial. So we suppose that $n\ge 3$.

We first prove that $K_{n}$ is minimal with respect to $\overline{\xi }^{c}$. If G is not a complete graph, then we can repeatedly add edges into G until we obtain $G\cong K_{n}$. By Lemma 2.1 (i), $\overline{\xi }^{c}(G)\ge \overline{\xi }^{c}(K_{n}),$ with equality if and only if $G\cong K_{n}$.

Now, let us assume that G is maximal with respect to $\overline{\xi }^{c}$. We shall prove that $G\cong P_{n}$. Suppose first that G is not isomorphic to a tree. Let Span(G) be one spanning tree of G. It then follows from Lemma 2.1 (ii) that $\overline{\xi }^{c}(G)<\overline{\xi }^{c}(Span(G))$, a contradiction to our choice of G. So, G must be a tree. We further claim that $G\cong P_{n}$. Suppose, to the contrary, that $G\ncong P_{n}$. Then, G has at least a branched vertex.

We choose a diametrical path, say $P_{d+1}:v_{0}v_{1}\ldots v_{d}$, in G. We claim that there exists no branched vertices outside the path $P_{d+1}$. If it is not so, we choose a branched vertex, say u, among all branched vertices outside the path $P_{d+1}$ such that $\max \{d_{G}(u,\,v_{0}),\,d_{G}(u,\,v_{d})\}=\max \Big \{\max \{d_{G}(x,\,v_{0}),\,d_{G}(x,\,v_{d})\}\Big \}$, where x is any one branched vertex in $V(G){\setminus } V(P_{d+1})$. Assume without loss of generality that $d_{G}(u,\,v_{d})=\max \{d_{G}(u,\,v_{0}),\,d_{G}(u,\,v_{d})\}$. Let $G-u=G_{1}\cup G_{2}\cup G_{3}\cdots \cup G_{k}$, where $G_{1}$ is assumed to be the component containing $v_{0}$ and $v_{d}$. Since u is a branched vertex, $k\ge 3$. By our choice of u, each $G_{i}$ ($i\ge 2$) can not have branched vertices, that is, each induced subgraph $G[G_{i}\cup \{u\}]\cong P_{n_i}$ for $i=2,\,\ldots ,\,k$. Let $u_{i}$ be another pendent vertex, different from u, of $P_{n_i}$ for $i=2,\,\ldots ,\,k$. Then, $d_{G}(u,\,v_{d})\ge d_{G}(u,\,u_{i})$ for each $i=2,\,\ldots ,\,k$, for otherwise, there exists some i such that $d_{G}(u_{i},\,v_{d})=d_{G}(u_{i},\,u)+d_{G}(u,\,v_{d})> 2d_{G}(u,\,v_{d})\ge d_{G}(u,\,v_{0})+d_{G}(u,\,v_{d})>d$, a contradiction.

Now, let $G_{0}=G\Big [\{u\}\cup \Big (V(G){\setminus } (V(G_{2})\cup V(G_{3}))\Big )\Big ]$. Then, $\varepsilon _{G_{0}}(u)\ge d_{G}(u,\,u_{d})\ge d_{G}(u,\,u_{2})$ and $\varepsilon _{G_{0}}(u)\ge d_{G}(u,\,u_{d})\ge d_{G}(u,\,u_{3})$ . So, we can employ Operation I, introduced as in Lemma 2.1, on G, and we get a new graph $G^{'}$. By Lemma 2.2, we have $\overline{\xi }^{c}(G)<\overline{\xi }^{c}(G^{'})$, a contradiction to our choice of G.

Similarly, the diametrical path $P_{d+1}$ cannot have branched vertices. If it is not so, we may choose a branched vertex, say u, among all branched vertices along the path $P_{d+1}:v_{0}v_{1}\ldots v_{d}$ such that $\max \{d_{G}(u,\,v_{0}),\,d_{G}(u,\,v_{d})\}=\max \Big \{\max \{d_{G}(x,\,v_{0}),\,d_{G}(x,\,v_{d})\}\Big \}$, where x is a branched vertex in $V(P_{d+1})$. Similar to above, we can employ Operation I on G to obtain a contradiction.

Therefore, $G\cong P_{n}$, as desired. $\square$

Remark 2.4

In fact, we may give a more direct proof than above for the first part of Theorem 2.3. According to (2), if a vertex is of degree $n-1$, then the contribution of this vertex to $\overline{\xi }^{c}$ is equal to 0. Since $K_{n}$ is the unique graph having the maximum number of vertices of degree $n-1$, $K_{n}$ is the unique graph minimal with respect to $\overline{\xi }^{c}$. But, Lemma 2.1 will be frequently used in the subsequent part of this paper. So, we use the current approach to prove the first part of Theorem 2.3.

3 Trees, Unicyclic Graphs, Bipartite Graphs Containing Cycles and Triangle-Free Graphs

In this section, we shall determine the tree, unicyclic graph, bipartite graph containing cycles and triangle-free graph with the minimum $\overline{\xi }^{c}$, respectively. First, we deduce a lower bound for $\overline{\xi }^{c}$ of a connected graph in terms of its order and size.

Theorem 3.1

Let G be a connected graph of order n, size m and diameter d. Then

$$\begin{aligned} \overline{\xi }^{c}(G)\ge 2n(n-1)-4m \end{aligned}$$

with equality if and only if $d\le 2$.

Proof

Suppose that N is the set of vertices of degree $n-1$, and $n_{0}$ is the number of elements in N. For any u in $V(G){\setminus } N$, we have $\varepsilon _{G}(u)\ge 2$. By (2), we have

$$\begin{aligned} \overline{\xi }^{c}(G)&=\sum \limits _{u\in V(G)}\varepsilon _{G}(u)( n-1-d_{G}(u))\\&=\sum \limits _{u\in N}\varepsilon _{G}(u)( n-1-d_{G}(u))+\sum \limits _{u\in V(G){\setminus } N}\varepsilon _{G}(u)( n-1-d_{G}(u))\\&\ge 2\sum \limits _{u\in V(G){\setminus } N}( n-1-d_{G}(u))\\&=2(n-1)(n-n_{0})-2\left( \sum \limits _{u\in V(G)}d_{G}(u)-n_{0}(n-1)\right) \\&=2n(n-1)-4m, \end{aligned}$$

where the equality is attained if and only if $\varepsilon _{G}(x)=2$ for each $x\in V(G){\setminus } N$, i.e., $\varepsilon _{G}(x)\le 2$ for each $x\in V(G)$, i.e., $d\le 2$.

This completes the proof. $\square$

According to Theorem 3.1, we get the following two results on $\overline{\xi }^{c}$ for trees and unicyclic graphs, respectively.

Corollary 3.2

Let T be a tree of order n. Then

$$\begin{aligned} \overline{\xi }^{c}(T)\ge 2n^{2}-6n+4 \end{aligned}$$

with equality if and only if $T\cong S_{n}$.

Proof

Suppose that $n_{0}$ is the number of vertices of degree $n-1$ in G. Then, $n_{0}=0 \text{ or } 1$. If $n_{0}=1$, then $G\cong S_{n}$, and $\overline{\xi }^{c}(G)=2n^{2}-6n+4$. Now, we assume that $n_{0}=0$. Let d be the diameter of G. Then, $d\ge 3$. By Theorem 3.1, $\overline{\xi }^{c}(G)>2n(n-1)-4m=2n(n-1)-4(n-1)=2n^{2}-6n+4$. This completes the proof. $\square$

Similarly, for a unicyclic graph, we have

Corollary 3.3

Let G be a unicyclic graph of order n. Then,

$$\begin{aligned} \overline{\xi }^{c}(G)\ge 2n^{2}-6n \end{aligned}$$

with equality if and only if $G\cong S_{n}^{3}$, where $S_{n}^{3}$ is the graph obtained by introducing an edge between two pendent vertices of the star $S_{n}$.

Now, we consider bipartite graphs containing cycles. We first prove a more general result which deals with the graphs with given chromatic number.

Theorem 3.4

Let G be a connected graph of order n with chromatic number $\chi$ such that $n=q\chi +p,\,0\le p\le \chi -1$. Then,

$$\begin{aligned} \overline{\xi }^{c}(G)\ge 4nq-2q(q+1)\chi \end{aligned}$$

with equality if and only if $G\cong T_{n,\,\chi }$.

Proof

Let $G_\mathrm{min}$ be a graph chosen among all connected graphs of order n with chromatic number $\chi$ such that $G_\mathrm{min}$ has the smallest $\overline{\xi }^{c}$. By Lemma 2.1(i), the addition of edges into a graph decreases its $\overline{\xi }^{c}$. Thus, we have $G_\mathrm{min}\cong \overline{K_{n_{1}}}\vee \overline{K_{n_{2}}}\vee \cdots \vee \overline{K_{n_{\chi }}}$, where $n_{i}$ is the number of vertices in the ith partite set.

By (2), we obtain

$$\begin{aligned} \overline{\xi }^{c}(G_\mathrm{min})&=\sum \limits _{i=1}^{\chi }n_{i}\cdot 2\Big [n-1-(n-n_{i})\Big ]\\&=\sum \limits _{i=1}^{\chi }2n_{i}(n_{i}-1)\\&=2\sum \limits _{i=1}^{\chi }n_{i}^{2}-2n. \end{aligned}$$

Suppose that $G_\mathrm{min}\ncong T_{n,\,\chi }$. Then, there exists $n_{j}\ge n_{i}+2$ for some $1\le i,\,j\le \chi$. We construct a new graph $G^{'}=\overline{K_{n_{1}}}\vee \cdots \overline{K_{n_{i}+1}}\vee \cdots \vee \overline{K_{n_{j}-1}}\vee \cdots \overline{K_{n_{\chi }}}$.

Then,

$$\begin{aligned} \overline{\xi }^{c}(G^{'})-\overline{\xi }^{c}(G_\mathrm{min})&=2[(n_{j}-1)^{2}-n_{j}^{2}+(n_{i}+1)^{2}-n_{i}^{2}]\\&=4(n_{i}+1-n_{j})\\&<0, \end{aligned}$$

a contradiction.

So, $G_\mathrm{min}\cong T_{n,\,\chi }$. Moreover, we have

$$\begin{aligned} \overline{\xi }^{c}(T_{n,\,\chi })&=p(q+1)\cdot 2\Big [n-1-(n-q-1)\Big ]+(\chi -p)q\cdot 2\Big [n-1-(n-q)\Big ]\\&=2pq(q+1)+2q(q-1)(\chi -p)\\&=4pq+2q(q-1)\chi \\&=4q(n-q\chi )+2q(q-1)\chi \\&=4nq-2q(q+1)\chi . \end{aligned}$$

This completes the proof. $\square$

Since a bipartite graph is a graph with chromatic number $\chi =2$, by Theorem 3.4, we have

Corollary 3.5

Let G be a cycle-containing bipartite graph of order n. Then,

$$\begin{aligned} \overline{\xi }^{c}(G)\ge \left\{ \begin{array}{ll} \displaystyle {n^{2}-2n+1} &{}\quad \text{ if } \text{ n } \text{ is } \text{ odd, } \\ \displaystyle {n^{2}-2n} &{}\quad \text{ if } \text{ n } \text{ is } \text{ even. } \end{array}\right. \end{aligned}$$

Each of above equalities holds if and only if $G\cong K_{\lfloor \frac{n}{2}\rfloor , \lceil \frac{n}{2}\rceil }$.

Now, we consider the triangle-free graph. First, we recall Turán’s Theorem, which is stated as follows.

Theorem 3.6

([26]) Let G be a connected $K_{q+1}$-free graph of order n and size m. Then,

$$\begin{aligned} m\le \Big \lfloor \Big (1-\frac{1}{q}\Big )\cdot \frac{n^{2}}{2}\Big \rfloor \end{aligned}$$

with equality if and only if G is a complete q-partite graph in which all classes are of almost equal cardinality.

Theorem 3.7

Let G be a connected triangle-free graph of order n. Then,

$$\begin{aligned} \overline{\xi }^{c}(G)\ge \left\{ \begin{array}{ll} \displaystyle {n^{2}-2n+1}&{}\quad \text{ if } \text{ n } \text{ is } \text{ odd, } \\ \displaystyle {n^{2}-2n}&{}\quad \text{ if } \text{ n } \text{ is } \text{ even. } \end{array}\right. \end{aligned}$$

(11)

Each of above equalities holds if and only if $G\cong K_{\lfloor \frac{n}{2}\rfloor , \lceil \frac{n}{2}\rceil }$.

Proof

Suppose that G is a connected triangle-free graph of order n and size m. Since G is triangle-free, by Theorem 3.6, we have

$$\begin{aligned} m\le & {} \Big \lfloor \Big (1-\frac{1}{2}\Big )\cdot \frac{n^{2}}{2}\Big \rfloor =\Big \lfloor \frac{n^{2}}{4}\Big \rfloor \end{aligned}$$

(12)

with equality holds if and only if $G\cong K_{\lfloor \frac{n}{2}\rfloor , \lceil \frac{n}{2}\rceil }$.

By Theorem 3.1 and (12),

$$\begin{aligned} \overline{\xi }^{c}(G)\ge & {} 2n(n-1)-4m\end{aligned}$$

(13)

$$\begin{aligned}\ge & {} 2n(n-1)-4\Big \lfloor \frac{n^{2}}{4}\Big \rfloor \nonumber \\= & {} \left\{ \begin{array}{l@{\quad }l} \displaystyle {n^{2}-2n+1} &{} \text{ if } n \text{ is } \text{ odd, } \\ \displaystyle {n^{2}-2n}&{} \text{ if } n \text{ is } \text{ even. } \end{array}\right. \end{aligned}$$

(14)

By Theorem 3.1, the equality in (13) holds if and only if $d\le 2$. From (12), we know that the equality in (14) holds if and only if $G\cong K_{\lfloor \frac{n}{2}\rfloor , \lceil \frac{n}{2}\rceil }$. So, the equality in (11) holds if and only if $G\cong K_{\lfloor \frac{n}{2}\rfloor , \lceil \frac{n}{2}\rceil }$.

This completes the proof. $\square$

4 Connected Graphs with Given Parameters

In this section, we will establish bounds for $\overline{\xi }^{c}$ of connected graphs with given parameters such as the number of pendent vertices, independence number, matching number, vertex-connectivity and edge-connectivity, respectively.

Theorem 4.1

Let G be a connected graph of order n with p pendent vertices. Then,

$$\begin{aligned} \overline{\xi }^{c}(G)\ge 4np-6p-2p^{2} \end{aligned}$$

with equality if and only if $G\cong K_{n}^{p}$.

Proof

Let $G_\mathrm{min}$ be a graph chosen among all connected graphs of order n with p pendent vertices such that $G_\mathrm{min}$ has the smallest $\overline{\xi }^{c}$. Let $v_{1},\,\ldots ,\,v_{p}$ be pendent vertices in $G_\mathrm{min}$. By Lemma 2.1(i), the addition of edges into a graph decreases its $\overline{\xi }^{c}$. So, the subgraph induced by vertices in $V(G_\mathrm{min}){\setminus }\{v_{1},\,\ldots ,\,v_{p}\}$ must be a complete subgraph in $G_\mathrm{min}$.

It is obvious that $G_\mathrm{min}$ has $p+\left( {\begin{array}{c}n-p\\ 2\end{array}}\right) =p+\frac{(n-p)(n-p-1)}{2}$ edges. By Theorem 3.1, we have

$$\begin{aligned} \nonumber \overline{\xi }^{c}(G_\mathrm{min})&\ge 2n(n-1)-4m\\ \nonumber&=2n(n-1)-4\Big [p+\frac{(n-p)(n-p-1)}{2}\Big ]\\ \nonumber&=2n(n-1)-4p-2(n-p)(n-p-1)\\&=4np-6p-2p^{2}. \end{aligned}$$

(15)

By the equality condition in Theorem 3.1, we know that the diameter of $G_\mathrm{min}$ must be equal to two. So, all pendent edges in $G_\mathrm{min}$ must be attached to the same vertex in $K_{n-p}$. Thus, the equality in (15) holds if and only if $G_\mathrm{min}\cong K_{n}^{p}$.

This completes the proof. $\square$

Theorem 4.2

Let G be a connected graph of order n with independence number $\alpha$. Then,

$$\begin{aligned} \overline{\xi }^{c}(G)\ge 2\alpha ^{2}-2\alpha \end{aligned}$$

with equality if and only if $G\cong \alpha K_{1}\vee K_{n-\alpha }$.

Proof

Let $G_\mathrm{min}$ be a graph chosen among all connected graphs of order n with independence number $\alpha$ such that $G_\mathrm{min}$ has the smallest $\overline{\xi }^{c}$. Let S be a maximal independent set in $G_\mathrm{min}$ with $|S|=\alpha$. Since adding edges into a graph will decrease its $\overline{\xi }^{c}$ by Lemma 2.1, each vertex u in S is adjacent to every vertex v in $G_\mathrm{min}-S$. Moreover, the subgraph induced by vertices in $G_\mathrm{min}-S$ is a complete subgraph of $G_\mathrm{min}$. So $G_\mathrm{min}\cong \alpha K_{1}\vee K_{n-\alpha }$. An elementary calculation gives $\overline{\xi }^{c}(\alpha K_{1}\vee K_{n-\alpha })=\alpha \cdot 2\Big [n-1-(n-\alpha )\Big ]=2\alpha ^{2}-2\alpha$, as claimed. $\square$

The following result on matching number is the well-known Tutte–Berge formula due to Tutte and Berge [6, 27].

Lemma 4.3

Suppose that G is a graph of order n with matching number $\beta$. Then,

$$\begin{aligned} n-2\beta = \max \{o(G -S)-|S| : S \subseteq V(G)\}, \end{aligned}$$

where o(G) denotes the number of odd components in G.

Theorem 4.4

Let G be a connected graph of order n with matching number $\beta \ge 1$.

(i):

If $\beta =\lfloor \frac{n}{2}\rfloor$, then

$$\begin{aligned} \overline{\xi }^{c}(G)\ge 0 \end{aligned}$$

with equality if and only if $G\cong K_{n}$.

(ii):

If $1\le \beta <\lfloor \frac{n}{2}\rfloor$, then

$$\begin{aligned} \overline{\xi }^{c}(G)\ge 2n^{2}-4n\beta +2\beta ^{2}-2n+2\beta \end{aligned}$$

with equality if and only if $G\cong K_{\beta }\vee (n-\beta )K_{1}$.

Proof

When $\beta =1$, we must have $G\cong K_{3}$ or $G\cong S_{n}$. If $n=2$, then $\beta =\lfloor \frac{2}{2}\rfloor$, and the result is obvious, as $G\cong S_{2}\cong K_{2}$. If $n=3$, then $\beta =\lfloor \frac{3}{2}\rfloor$. By Lemma 2.1, it is easy to check that $\overline{\xi }^{c}(S_{3})>\overline{\xi }^{c}(K_{3})$, and the result follows readily. If $n\ge 4$, then $\beta =1<\lfloor \frac{4}{2}\rfloor$. Since $S_{n}\cong K_{\beta }\vee (n-\beta )K_{1}$ for $\beta =1$, the result holds.

Now, we assume that $\beta \ge 2$, and then $n\ge 4$.

We choose $G_\mathrm{min}$ to be a graph such that $G_\mathrm{min}$ has the smallest $\overline{\xi }^{c}$ among all connected graphs of order n with matching number $\beta$. According to Lemma 4.3, there exists a vertex subset S, satisfying $|S|=s$, in $V(G_\mathrm{min})$ such that $G_\mathrm{min}-S$ has $t=n-2\beta +s$ odd components, say $G_{1},\,\ldots ,\,G_{t}$. For each $i=1,\,\ldots ,\,t$, let $n_{i}$ be the order of $G_{i}$. Then, each $n_i$ is a positive odd number for $i=1,\,\ldots ,\,t$ and $\sum \nolimits _{i=1}^{t}n_{i}=n-s$. $\square$

We have the following claim.

Claim 1

$G_\mathrm{min}\cong K_{s}\vee \Big (\bigcup \limits _{i=1}^{t}K_{n_{i}}\Big )$.

Proof

Assume without loss of generality that $n_{1}\le n_{2}\le \ldots \le n_{t}$. We first show that $G_\mathrm{min}-S$ contains no even components. If it is not so, we may let U be the union of all even components of $G_\mathrm{min}-S$. Now, one can add all possible edges between vertices in U and those in $G_{t}$, until the resulting subgraph induced by vertices both in U and in $G_{t}$ is a complete subgraph. The resulting graph obtained from $G_\mathrm{min}$ by adding edges in such a way as above is denoted by $G^{*}$. By Lemma 4.3, on one hand, we have

$$\begin{aligned} n-2\beta (G^{*}) \ge o(G^{*} -S)-|S| =o(G -S)-|S|=n-2\beta (G) , \end{aligned}$$

implying that $\beta (G^{*}) \le \beta (G)$. On the other hand, we have $\beta (G^{*}) \ge \beta (G)$. Thus, $\beta (G^{*}) =\beta (G)$. But then, by Lemma 2.1, we have $\overline{\xi }^{c}(G_\mathrm{min})>\overline{\xi }^{c}(G^{*})$, a contradiction to our choice of $G_\mathrm{min}$. So, all components of $G_\mathrm{min}-S$ are odd and thus, $G_\mathrm{min}-S=G_{1}\cup \cdots \cup G_{t}$. It is not difficult to see that each $G_{i}$ is a complete subgraph, for otherwise, we can add edges into any one non-complete subgraph, say $G_{j}$, and we obtain a new graph $G^{**}$ of order n. Similar to above, we have $\beta (G^{**}) =\beta (G)$. Again, by Lemma 2.1, we have $\overline{\xi }^{c}(G_\mathrm{min})>\overline{\xi }^{c}(G^{**})$, a contradiction. Similarly, we can prove that, for each $i=1,\ldots ,\,t$, $G[V(G_{i})\cup S]$ is a complete subgraph of $G_\mathrm{min}$. So, $G_\mathrm{min}\cong K_{s}\vee \Big (\bigcup \nolimits _{i=1}^{t}K_{n_{i}}\Big )$. $\square$

By Claim 1, there exists a vertex subset S having s vertices in $G_\mathrm{min}$ such that $G_\mathrm{min}\cong K_{s}\vee \Big (\bigcup \nolimits _{i=1}^{t}K_{n_{i}}\Big )$, where each $n_i$ is a positive odd number for $i=1,\,\ldots ,\,t$.

If $s=0$, then $n-2\beta =t$. Since $G_\mathrm{min}$ is connected, we have $t\le 1$. If $t=0$, then $n=2\beta$; If $t=1$, then $n=2\beta +1$. So, when $s=0$, we have $\beta =\lfloor \frac{n}{2}\rfloor$. When $\beta =\lfloor \frac{n}{2}\rfloor$, we conclude that $G_\mathrm{min}\cong K_{n}$, for otherwise, we can add edges into $G_\mathrm{min}$ so that we can obtain a new graph with strictly smaller $\overline{\xi }^{c}$ that that of $G_\mathrm{min}$, a contradiction.

Now, we assume that $s\ge 1$ and $\beta <\lfloor \frac{n}{2}\rfloor$.

By (2), we have $\overline{\xi }^{c}\Big ( K_{s}\vee \Big (\bigcup \nolimits _{i=1}^{t}K_{n_{i}}\Big )\Big )= \sum \nolimits _{i=1}^{t}2n_{i}[n-(n_{i}-1+s)-1]=2(n-s)^{2}-2\sum \nolimits _{i=1}^{t}n_{i}^{2}$. We claim that $n_{1}=\ldots =n_{t-1}=1$ and $n_{t}=n-s-t+1$, that is, $G_\mathrm{min}\cong K_{s}\vee \Big (K_{n-s-t+1}\bigcup (t-1)K_{1}\Big )$. Suppose to the contrary that $n_{j}\ge 3$ for some $j\in \{1,\,2,\,\ldots ,\,t-1\}$. Let $G^{***}=K_{s}\vee \Big (K_{n_{t}+2}\bigcup K_{n_{j}-2}\bigcup \Big (\bigcup \nolimits _{i=1,\,i\ne j}^{t-1}K_{n_{i}}\Big )\Big )$. Clearly, $o(G^{***} -S)=o(G_\mathrm{min} -S)$, and thus, $\beta (G^{***})=\beta (G_\mathrm{min})$.

But then, we have

$$\begin{aligned} \overline{\xi }^{c}(G_\mathrm{min})- \overline{\xi }^{c}(G^{***})&=-2(n_{j}^{2}+n_{t}^{2})+2[(n_{j}-2)^{2}+(n_{t}+2)^{2}]\\&=8(-n_{j}+n_{t}+2)>0, \end{aligned}$$

a contradiction to our choice of $G_\mathrm{min}$.

Thus, $n_{1}=\ldots =n_{t-1}=1$ and $n_{t}=n-s-t+1$, that is, $G_\mathrm{min}\cong K_{s}\vee \Big (K_{n-s-t+1}\bigcup (t-1)K_{1}\Big )$. By (2), $\overline{\xi }^{c}\Big ( K_{s}\vee \Big (K_{n-s-t+1}\bigcup (t-1)K_{1}\Big )\Big )=2(t-1)(2n-2s-t)$. Let $f(x)=2(t-1)(2n-2x-t)$. Clearly $t\ge 2$. Then, f(x) is a strictly decreasing function on the interval $[1,\,\beta ]$.

Since each $n_{i}\ge 1$, we have $n\ge s+t$. So, $n\ge n+2s-2\beta$, resulting in $s\le \beta$. When $s=\beta$, we have $n-s-t+1=n-\beta -t+1$. Recall that $n-2\beta =t-s=t-\beta$, implying that $t=n-\beta$. Thus, $n-s-t+1=1$. If $s<\beta$, then $f(s)>f(\beta )$, that is, $\overline{\xi }^{c}(G_\mathrm{min})=\overline{\xi }^{c}\Big ( K_{s}\vee \Big (K_{n-s-t+1}\bigcup (t-1)K_{1}\Big )\Big )>\overline{\xi }^{c}\Big ( K_{\beta }\vee (n-\beta )K_{1}\Big )$, a contradiction to our choice of $G_\mathrm{min}$.

So, $s=\beta$, and $G_\mathrm{min}\cong K_{\beta }\vee (n-\beta )K_{1}$.

This completes the proof. $\square$

In the following two theorems, we shall determine graphs with the minimum $\overline{\xi }^{c}$ among graphs with given vertex-connectivity and edge-connectivity, respectively.

Theorem 4.5

Let G be a graph of order n with vertex-connectivity $\kappa$. Then,

$$\begin{aligned} \overline{\xi }^{c}(G)\ge 4(n-\kappa -1) \end{aligned}$$

with equality if and only if $G\cong K_{\kappa }\vee (K_{1}+K_{n-\kappa -1})$.

Proof

We choose $G_\mathrm{min}$ to be a graph such that $G_\mathrm{min}$ has the smallest $\overline{\xi }^{c}$ within all connected graphs of order n with vertex-connectivity $\kappa$. Let C be a vertex-cut in $G_\mathrm{min}$ such that $|C|=\kappa$ and let $G_\mathrm{min}-C=G_{1}\cup G_{2}\cup \cdots G_{t}\,(t\ge 2)$. By Lemma 2.1, we must have $t=2$, for otherwise, we can add edges between any two components, resulting in a new graph $G^{*}$ with vertex-connectivity $\kappa$ and a strictly smaller $\overline{\xi }^{c}$ than that of $G_\mathrm{min}$, a contradiction to our choice of $G_\mathrm{min}$.

By the same reason, we can deduce that both $G_{1}$ and $G_{2}$ are cliques of $G_\mathrm{min}$, that the subgraph of $G_\mathrm{min}$ induced by C is a clique, and that any vertex in $G_{1}\cup G_{2}$ is adjacent to each vertex in C. Let $n_{i}$ denote the order of $G_{i}$. Thus, we have $G_\mathrm{min}\cong K_{\kappa }\vee (K_{n_{1}}+K_{n_{2}})$.

Without loss of generality, we may assume that $n_{2}\ge n_{1}$. If $n_{1}=1$, then the theorem follows. Suppose now that $n_{1}\ge 2$. By (2), we obtain

$$\begin{aligned} \overline{\xi }^{c}(G_\mathrm{min})&=\sum \limits _{u\in V(G_{1})}\varepsilon _{G_\mathrm{min}}(u)\Big (n-1-d_{G_\mathrm{min}}(u)\Big )\\&\quad +\sum \limits _{u\in V(G_{2})}\varepsilon _{G_\mathrm{min}}(u)\Big (n-1-d_{G_\mathrm{min}}(u)\Big )\\&\quad +\sum \limits _{u\in C}\varepsilon _{G_\mathrm{min}}(u)\Big (n-1-d_{G_\mathrm{min}}(u)\Big )\\&=4n_{1}n_{2}. \end{aligned}$$

Let $G^{*}=K_{\kappa } \vee (K_{n_{1}-1}+K_{n_{2}+1})$. Then

$$\begin{aligned} \overline{\xi }^{c}(G^{*})- \overline{\xi }^{c}(G_\mathrm{min})&= 4\Big [(n_{1}-1)(n_{2}+1)-n_{1}n_{2}\Big ]\\&=4(n_{1}-n_{2}-1)<0, \end{aligned}$$

a contradiction to our choice of $G_\mathrm{min}$.

So, $n_{1}=1$ and $n_{2}=n-\kappa -1$. Thus, $G_\mathrm{min}\cong K_{\kappa }\vee (K_{1}+K_{n-\kappa -1})$.

An elementary calculation gives $\overline{\xi }^{c}(K_{\kappa }\vee (K_{1}+K_{n-\kappa -1}))= 4(n-\kappa -1)$, completing the proof. $\square$

In our last theorem, we determine the graph with the minimum $\overline{\xi }^{c}$ among all graphs of order n with edge-connectivity $\kappa ^{'}$.

Theorem 4.6

Let G be a graph of order n with edge-connectivity $\kappa ^{'}$. Then

$$\begin{aligned} \overline{\xi }^{c}(G)\ge 4(n-\kappa ^{'}-1) \end{aligned}$$

with equality if and only if $G\cong K_{\kappa ^{'}}\vee (K_{1}+K_{n-\kappa ^{'}-1})$.

Proof

Let $g(x)=4(n-x-1)$. It is easily seen that g(x) is a strictly decreasing function. Suppose that G is a graph of order n with vertex-connectivity $\kappa$ and edge-connectivity $\kappa ^{'}$. Then, $\kappa \le \kappa ^{'}$. It follows from Theorem 4.5 that $\overline{\xi }^{c}(G)\ge g(\kappa )$. Since $g(\kappa )\ge g(\kappa ^{'})$, we get $\overline{\xi }^{c}(G)\ge g(\kappa ^{'})=4(n-\kappa ^{'}-1)$. It is easy to check that the equality holds if and only if $G\cong K_{\kappa ^{'}}\vee (K_{1}\cup K_{n-1-\kappa ^{'}})$.

This completes the proof. $\square$

5 Concluding Remarks

In this paper, we considered a new eccentricity-based graph invariant, named the eccentric connectivity coindex. We mainly investigated extremal properties of this graph invariant. More specifically, we characterized extremal graphs with the maximum and minimum $\overline{\xi }^c$, respectively, among all connected graphs of given order. Also, we characterized the connected graph with given order, size and the minimum $\overline{\xi }^{c}$ as well as the tree, unicyclic graph, bipartite graph containing cycles and triangle-free graph with the minimum $\overline{\xi }^{c}$, respectively. Moreover, we established various lower bounds for $\overline{\xi }^{c}$ in terms of several other graph parameters including the number of pendent vertices, independence number, matching number, chromatic number, vertex-connectivity, and edge-connectivity.

Our research on this new graph invariant is just a beginning. Similar to other distance-based invariants, there are many interesting problems about this graph invariant left for us to discover and solve in the future.