On the geometric–arithmetic index by decompositions-CMMSE

Abstract

The concept of geometric–arithmetic index was introduced in the chemical graph theory recently, but it has shown to be useful. There are many papers studying different kinds of indices (as Wiener, hyper–Wiener, detour, hyper–detour, Szeged, edge–Szeged, PI, vertex–PI and eccentric connectivity indices) under particular cases of decompositions. The main aim of this paper is to show that the computation of the geometric-arithmetic index of a graph G is essentially reduced to the computation of the geometric-arithmetic indices of the so-called primary subgraphs obtained by a general decomposition of G. Furthermore, using these results, we obtain formulas for the geometric-arithmetic indices of bridge graphs and other classes of graphs, like bouquet of graphs and circle graphs. These results are applied to the computation of the geometric-arithmetic index of Spiro chain of hexagons, polyphenylenes and polyethene.

1 Introduction

A topological index is defined as a single number that represents a chemical structure in graph-theoretical terms via the molecular graph, and that correlates with a molecular property; it is used to understand physicochemical properties of chemical compounds. Topological indices are interesting since they capture some of the properties of a molecule in a single number. Hundreds of topological indices have been introduced and studied, starting with the seminal work by Wiener [25] in which he used the sum of all shortest-path distances of a (molecular) graph for modeling physical properties of alkanes.

Topological indices based on end-vertex degrees of edges have been used over 40 years. Among them, several indices are recognized to be useful tools in chemical researches. Probably, the best know such descriptor is the Randić connectivity index (R) [15]. There are more than thousand papers and a couple of books dealing with this index (see e.g., [7, 8, 16] and the references therein). During many years, scientists were trying to improve the predictive power of the Randić index. This led to the introduction of a large number of new topological descriptors resembling the original Randić index. The first geometric-arithmetic index $GA _1$, defined in [24] as

$$\begin{aligned} GA _1(G) = \sum _{uv\in E(G)}\frac{\sqrt{d_u d_v}}{\frac{1}{2} (d_u + d_v)} \end{aligned}$$

where uv denotes the edge of the graph G connecting the vertices u and v, and $d_u$ is the degree of the vertex u, is one of the successors of the Randić index. Although $GA _1$ was introduced just six years ago, there are many papers dealing with this index (see e.g., [5, 13, 17, 18, 21, 24] and the survey [4]). There are other geometric-arithmetic indices, like $Z_{p,q}$ ($Z_{0,1} = GA _1$), but the results in [4, p. 598] show empirically that the $GA _1$ index gathers the same information on observed molecules as other $Z_{p,q}$ indices.

The reason for introducing a new index is to gain prediction of some property of molecules somewhat better than obtained by already presented indices. Therefore, a test study of predictive power of a new index must be done. As a standard for testing new topological descriptors, the properties of octanes are commonly used. We can find 16 physico-chemical properties of octanes at www.moleculardescriptors.eu.

Although only about 1000 benzenoid hydrocarbons are known, the number of possible benzenoid hydrocarbons is huge. For instance, the number of possible benzenoid hydrocarbons with 35 benzene rings is 5851000265625801806530 [23]. Therefore, the modeling of their physico-chemical properties is very important in order to predict properties of currently unknown species.

The graphic in [4, Fig. 7] (from [4, Table 2], [22]) shows that there exists a good linear correlation between $GA _1$ and the heat of formation of benzenoid hydrocarbons (the correlation coefficient is equal to 0.972). Furthermore, the improvement in prediction with $GA _1$ index comparing to Randić index in the case of standard enthalpy of vaporization is more than 9 $\%$. Hence, one can think that $GA _1$ index should be considered in the QSPR/QSAR researches.

We say that a family of subgraphs $\{G_1,\dots ,G_r\}$ of G is a decomposition of G if $G_1\cup \cdots \cup G_r = G$ and $G_i \cap G_j$ is either empty or a vertex for every $i,j \in \{1,\dots ,r\} \,, i\ne j$. The subgraphs $G_1,\dots ,G_r$ are usually called primary subgraphs of the decomposition.

Particular cases of decompositions are the T-decompositions (equivalent to the concept of graphs obtained by point-attaching, see [6, 19]); they are a very useful tool in different areas of graph theory, as Gromov hyperbolic graphs (see e.g., [2, 3, 12]), polynomials on graphs (see [6]) and metric dimension of graphs (see [19]). We say that a vertex v of a graph G is a cut-vertex if $G {\setminus } \{v\}$ is not connected. A graph is biconnected if it does not contain cut-vertices. Given a graph G, we say that a family of subgraphs $\{G_1,\dots ,G_r\}$ of G is a T-decomposition of G if $G_1\cup \cdots \cup G_r = G$ and $G_i \cap G_j$ is either empty or a cut-vertex for every $i,j \in \{1,\dots ,r\} \,, i\ne j$. Every graph has a T-decomposition, as the following example shows. Given any edge in G, let us consider the maximal two-connected subgraph containing it: this is the well-known biconnected decomposition of G.

In [9] and [10] the authors introduce the concept of bridge and chain graphs, which are particular cases of T-decompositions (a chain graph is a graph with a T-decomposition in which every cut-vertex belongs at most to two primary subgraphs; bridge graphs are a subset of chain graphs). For bridge and chain graphs the PI index was determined in [9] and for bridge graphs the Szeged index (and the vertex PI index) was considered in [11]. Recently, the Wiener index was considered in [1] for a class of graphs containing as special cases the bridge and chain graphs. Also, in [10] appear formulas for the Wiener, hyper–Wiener, detour and hyper–detour indices of bridge and chain graphs. Besides, [20] contains formulas for the Szeged, edge–Szeged, PI, vertex–PI and eccentric connectivity indices of splice graphs (another class of graphs with T-decompositions). More classes of graphs can be obtained as particular cases of T-decompositions: bouquets of graphs, rooted product graphs, corona product graphs and block graphs (see [19]).

Since the papers [1, 9–11, 20] (studying different kinds of indices under particular cases of decompositions) are useful, it is natural to show that the computation of the geometric-arithmetic index of a graph G is essentially reduced to the computation of the geometric-arithmetic indices of the primary subgraphs obtained by a general decomposition of G. This is the main aim of the present paper. Using these results, we obtain formulas for the geometric-arithmetic indices of bridge graphs and other classes of graphs, like bouquet of graphs and circle graphs. These results are applied in Sect. 3 to the computation of the geometric-arithmetic index of Spiro chain of hexagons, Polyphenylenes and Polyethene (see Examples 3.2, 3.4 and 3.8, respectively).

Throughout this paper, $G=(V (G),E (G))$ denotes a (non-oriented) finite simple (without multiple edges and loops) connected graph with $E(G) \ne \emptyset$. Note that the connectivity of G is not an important restriction, since if G has connected components $G_1,\dots ,G_r,$ then $GA _1(G) = GA _1(G_1) + \cdots + GA _1(G_r)$; furthermore, every molecular graph is connected.

2 Geometric–arithmetic index and decompositions

If $v\in V(G)$, then we denote by $N_G(v)$ or N(v) the set of neighbors of v, i.e.,

$$\begin{aligned} N_G(v)=N(v)=\big \{u\in V(G)\,|\; uv\in E(G) \big \}. \end{aligned}$$

Given a decomposition $\{G_1,\dots ,G_r\}$ of G, denote by $\mathcal {W}$ the set of vertices v in G belonging at least at two $G_i$’s. Given a vertex $v \in \mathcal {W}$, denote by $G_{i_1},\dots ,G_{i_k}$ the set of primary subgraphs containing v and by $d_{i_j}$ the number of neighbors of v in $G_{i_j}$ (then $d_v=d_{i_1}+\dots + d_{i_k}$). If $v \in \mathcal {W}$, then we define W(v) as

$$\begin{aligned} W(v) = \sum _{u\in N_{G}(v) {\setminus } \mathcal {W}}\frac{\sqrt{d_u d_v}}{\frac{1}{2} (d_u + d_v)} - \sum _{j=1}^k \sum _{u\in N_{G_{i_j}}(v) {\setminus } \mathcal {W}} \frac{\sqrt{d_u d_{i_j}}}{\frac{1}{2} (d_u + d_{i_j})}. \end{aligned}$$

Denote by $\mathcal {Z}$ the set of edges in G with both endpoints in $\mathcal {W}$. If $e=uv \in \mathcal {Z}$, then $e \in G_i$ for some i, and we denote by $d_u^*,d_v^*$ the degrees of u, v in $G_i$. If $e=uv \in \mathcal {Z}$, then we define Z(e) as

$$\begin{aligned} Z(e) = \frac{\sqrt{d_u d_v}}{\frac{1}{2} (d_u + d_v)} - \frac{\sqrt{d_u^* d_v^*}}{\frac{1}{2} (d_u^* + d_v^*)}. \end{aligned}$$

The following result allows to compute the precise value of $GA _1(G)$ in terms of the geometric-arithmetic indices of the primary subgraphs in any decomposition.

Theorem 2.1

Let $\{G_1,\dots ,G_r\}$ be a decomposition of the graph G. Then

$$\begin{aligned} GA _1(G) = \sum _{i=1}^r GA _1(G_i) + \sum _{v \in \mathcal {W}} W(v) + \sum _{e \in \mathcal {Z}} Z(e) . \end{aligned}$$

Proof

First of all, note that if $u,v \notin \mathcal {W}$ and $uv\in E(G)$, then the term in $GA _1(G)$ corresponding to uv in G is equal to its corresponding term in $\sum _{i=1}^r GA _1(G_i)$.

For each $v \in \mathcal {W}$ and $u \notin \mathcal {W}$ with $uv\in E(G)$, W(v) replaces in the sum $\sum _{i=1}^r GA _1(G_i)$ the corresponding term to uv by its correct value as edge in G. This fact holds since the degree of u is $d_u$ both in G and in its (unique) corresponding primary subgraph.

Finally, for each $u,v \in \mathcal {W}$ with $uv\in E(G)$, Z(uv) replaces in the sum $\sum _{i=1}^r GA _1(G_i)$ the corresponding term to uv by its correct value as edge in G. $\square$

In order to estimate the difference between $GA _1(G)$ and $\sum _{i=1}^r GA _1(G_i)$, Proposition 2.4 will provide bounds for W(v) and Z(uv). We need first the following technical result.

Lemma 2.2

Given any positive integer a, let $h_{a}$ be the function $h_{a}(x)=\frac{2\sqrt{x}}{a +x}$ on the interval $[1,\infty )$. Then $h_{a}$ strictly increases in [1, a], strictly decreases in $[a,\infty )$, and $|h_{a}'(x)| \le \frac{1}{8}$ for every $x\in [1,\infty )$.

Proof

Since

$$\begin{aligned} h_{a}'(x) =\frac{ \frac{1}{\sqrt{x}} (a+x) - 2\sqrt{x}}{(a +x)^2} =\frac{a-x}{\sqrt{x}\,(a +x)^2} \,, \end{aligned}$$

$h_{a}' > 0$ on [1, a) and $h_{a}' < 0$ on $(a,\infty )$. A computation gives

$$\begin{aligned} h_{a}''(x)= & {} \frac{ -\sqrt{x}\,(a +x)^2 +(x-a)\Big ( \frac{1}{2\sqrt{x}} (a+x)^2 + 2 \sqrt{x} \,(a +x) \Big )}{x(a +x)^4} \\= & {} \frac{ 3x^2-6ax-a^2}{2\,x^{3/2} (a +x)^3}. \end{aligned}$$

We have $3x^2-6ax-a^2=0$ if and only if

$$\begin{aligned} x = \frac{3a \pm \sqrt{9a^2+3a^2}}{3} = \frac{3 \pm 2\sqrt{3}}{3}\,a. \end{aligned}$$

Thus, $h_{a}'' \le 0$ on $\big [1,\frac{3 + 2\sqrt{3}}{3}\,a\big ]$ and $h_{a}'' \ge 0$ on $\big [\frac{3 + 2\sqrt{3}}{3}\,,\infty )$. Since $\lim _{x\rightarrow \infty } h_{a}'(x) = 0$ and $a\ge 1$,

$$\begin{aligned} |h_{a}'(x)| \le \max \left\{ h_{a}'(1), -h_{a}'\left( \frac{3 + 2\sqrt{3}}{3}\,a\right) \, \right\} = \max \left\{ \frac{a-1}{(a +1)^2}\,, \frac{c_0}{a^{3/2}}\, \right\} \end{aligned}$$

for every $x\in [1,\infty )$, where

$$\begin{aligned} c_0=\frac{\frac{2\sqrt{3}}{3}}{\sqrt{\frac{3 + 2\sqrt{3}}{3}}\,\big (\frac{6 + 2\sqrt{3}}{3}\big )^2}\sim 0.079 . \end{aligned}$$

Consider the function

$$\begin{aligned} H(t) =\frac{t^{3/2}(t-1)}{(t +1)^2} =\frac{t^{5/2}-t^{3/2}}{(t +1)^2} \end{aligned}$$

for $t \in [2,\infty )$. Since

$$\begin{aligned} H'(t) =\frac{\big (\frac{5}{2} \,t^{3/2}-\frac{3}{2} \,t^{1/2}\big )(t +1)^2 -\big (t^{5/2}-t^{3/2}\big )2(t +1)}{(t +1)^4} =\frac{t^{1/2}}{2}\,\frac{t^2 +6t-3}{(t +1)^3} >0 \end{aligned}$$

for $t \in [2,\infty )$, we have

$$\begin{aligned} \frac{t^{3/2}(t-1)}{(t +1)^2} = H(t) \ge H(2) = \frac{2^{3/2}}{9}> c_0, \quad \frac{t-1}{(t +1)^2} > \frac{c_0}{t^{3/2}}\,, \end{aligned}$$

for every $t \in [2,\infty )$. Therefore,

$$\begin{aligned} |h_{a}'(x)| \le \max \left\{ \frac{a-1}{(a +1)^2}\,, \frac{c_0}{a^{3/2}}\, \right\} =\left\{ \begin{array}{ll} c_0, &{}\quad \text{ if } \ a=1,\\ \frac{a-1}{(a +1)^2}, &{}\quad \text{ if } \ a \ge 2. \end{array} \right. \end{aligned}$$

for every $x\in [1,\infty )$. Since $u(t)=\frac{t-1}{(t +1)^2}$ satisfies

$$\begin{aligned} u'(t) =\frac{(t +1)^2-(t -1)2(t +1)}{(t +1)^4} =\frac{3-t}{(t +1)^3} \,, \end{aligned}$$

we obtain $u'(t)\ge 0$ for every $t\in [2,3]$ and $u'(t)\le 0$ for every $t\in [3,\infty )$. Hence, $u(a)\le u(3)= \frac{1}{8} > c_0$ for every $a\ge 2$ and $|h_{a}'(x)| \le \frac{1}{8}$ for every $x\in [1,\infty )$. $\square$

We also need the following result (see e.g., [17, Corollary 2.3]).

Lemma 2.3

Let g be the function $g(x,y)=\frac{2\sqrt{xy}}{x + y}$ with $0<a\le x,y \le b$. Then

$$\begin{aligned} \frac{2\sqrt{ab}}{a + b} \le g(x,y) \le 1. \end{aligned}$$

The equality in the lower bound is attained if and only if either $x=a$ and $y=b$, or $x=b$ and $y=a$, and the equality in the upper bound is attained if and only if $x=y$. Besides, $g(x,y)= g(x',y')$ if and only if x / y is equal to either $x'/y'$ or $y'/x'$.

Given a decomposition $\{G_1,\dots ,G_r\}$ of G and $e=uv \in \mathcal {Z}$, we say that e is maximal or minimal if $d_u =d_v$ or $d_u^* =d_v^*$, respectively.

Given a graph G, denote by $\Delta ,\delta$ the maximum and minimum degrees of G, respectively.

Proposition 2.4

Let $\{G_1,\dots ,G_r\}$ be a decomposition of the graph G. Given $e \in \mathcal {Z}$, denote by $\Delta _e,\delta _e$ the maximum and minimum degrees of the primary subgraph $G_i$ with $e\in G_i$, respectively. Then

$$\begin{aligned} -1 \le \frac{2\sqrt{\Delta \delta }}{\Delta + \delta } - 1 \le Z(e) \le 1 - \frac{2\sqrt{\Delta _e\delta _e}}{\Delta _e + \delta _e} \le 1 , \end{aligned}$$

for every $e \in \mathcal {Z}$. If e is maximal or minimal, then $Z(e)\ge 0$ or $Z(e) \le 0$, respectively. Furthermore,

$$\begin{aligned} |W(v)| \le \frac{1}{8}\, \sqrt{\Delta } \,\, d_{v}(d_{v}-1) , \end{aligned}$$

for every $v \in \mathcal {W}$.

Proof

If $e=uv \in \mathcal {Z}$, then Lemma 2.3 gives

$$\begin{aligned} -1 \le \frac{2\sqrt{\Delta \delta }}{\Delta + \delta } - 1 \le Z(e) = \frac{\sqrt{d_u d_v}}{\frac{1}{2} (d_u + d_v)} - \frac{\sqrt{d_u^* d_v^*}}{\frac{1}{2} (d_u^* + d_v^*)} \le 1 - \frac{2\sqrt{\Delta _e\delta _e}}{\Delta _e + \delta _e} \le 1. \end{aligned}$$

If e is maximal or minimal, then

$$\begin{aligned} Z(e) = \frac{\sqrt{d_u d_v}}{\frac{1}{2} (d_u + d_v)} - \frac{\sqrt{d_u^* d_v^*}}{\frac{1}{2} (d_u^* + d_v^*)} = 1 - \frac{\sqrt{d_u^* d_v^*}}{\frac{1}{2} (d_u^* + d_v^*)} \ge 0 \end{aligned}$$

   or    

$$\begin{aligned} Z(e) = \frac{\sqrt{d_u d_v}}{\frac{1}{2} (d_u + d_v)}- 1 \le 0, \end{aligned}$$

respectively.

Finally, fix $v \in \mathcal {W}$. As in the definition of W(v), denote by $G_{i_1},\dots ,G_{i_k}$ the set of primary subgraphs containing v and by $d_{i_j}$ the number of neighbors of v in $G_{i_j}$ for $j=1,\dots ,k$. Fix j and $u \in N_{G_{i_j}}(v) {\setminus } \mathcal {W}$. Mean value theorem and Lemma 2.2 give

$$\begin{aligned} \left| \, \frac{2\sqrt{d_u d_v}}{d_u +d_v} - \frac{2\sqrt{d_u d_{i_j}}}{d_u +d_{i_j}} \, \right| = \sqrt{d_u} \,\, \big | \, h_{d_u}(d_v) - h_{d_u}(d_{i_j}) \big | = \sqrt{d_u} \,\, \big | \, h_{d_u}'(t) \big | \,\big (d_v-d_{i_j}\big ) \end{aligned}$$

for some $t \in (d_{i_j},d_v) \subset [1,d_v]$, where $h_a$ is the function appearing in Lemma 2.2. Since $| h_{d_u}'(t) | \le \frac{1}{8}$ for every $t\in [1,\infty )$ by Lemma 2.2, we deduce

$$\begin{aligned} \left| \, \frac{2\sqrt{d_u d_{i_j}}}{d_u +d_{i_j}} - \frac{2\sqrt{d_u d_v}}{d_u +d_v} \, \right| \le \sqrt{\Delta } \,\, \frac{1}{8} \,\big (d_v-1\big ) . \end{aligned}$$

Hence,

$$\begin{aligned} |W(v)| \le \frac{1}{8}\, \sqrt{\Delta } \,\, d_{v}(d_{v}-1) , \end{aligned}$$

for every $v \in \mathcal {W}$. $\square$

We have the following direct consequence of Theorem 2.1, since $|Z(e)| \le 1$ for every $e \in \mathcal {Z}$ by Proposition 2.4.

Corollary 2.5

Let $\{G_1,\dots ,G_r\}$ be a decomposition of the graph G. Then

$$\begin{aligned} \left| \, GA _1(G) - \sum _{i=1}^r GA _1(G_i) \, \right| \le \frac{1}{8}\, \Delta ^{3/2}(\Delta -1) \,card\, \mathcal {W} + card\, \mathcal {Z} . \end{aligned}$$

Proposition 2.6

Let $\{G_1,\dots ,G_r\}$ be a decomposition of the graph G. If $d_{v}\le d_u$ for every $v\in \mathcal {W}$ and $u \in N_{G}(v) {\setminus } \mathcal {W}$, then

$$\begin{aligned} GA _1(G) \ge \sum _{i=1}^r GA _1(G_i) - card\, \mathcal {Z} . \end{aligned}$$

Furthermore, if every edge in $\mathcal {Z}$ is maximal, then

$$\begin{aligned} GA _1(G) \ge \sum _{i=1}^r GA _1(G_i) . \end{aligned}$$

Proof

Fix $v \in \mathcal {W}$. We are going to prove $W(v) \ge 0$. Denote by $G_{i_1},\dots ,G_{i_k}$ the set of primary subgraphs containing v and by $d_{i_j}$ the number of neighbors of v in $G_{i_j}$, as in the definition of W(v). Fix j and $u \in N_{G_{i_j}}(v) {\setminus } \mathcal {W}$. Since

$$\begin{aligned} h_{d_u}(x)=\frac{2\sqrt{x}}{d_u +x} \end{aligned}$$

is an increasing function on $[1,d_u]$ by Lemma 2.2, and $d_{i_j} \le d_{v} \le d_u$,

$$\begin{aligned} \frac{\sqrt{d_u d_{i_j}}}{\frac{1}{2} (d_u + d_{i_j})} \le \frac{\sqrt{d_u d_v}}{\frac{1}{2} (d_u + d_v)}. \end{aligned}$$

Hence, $W(v) \ge 0$ for every $v \in \mathcal {W}$. Since $|Z(e)| \le 1$ for every $e \in \mathcal {Z}$ by Proposition 2.4, Theorem 2.1 gives the result.

Furthermore, if every edge in $\mathcal {Z}$ is maximal, then Proposition 2.4 gives $Z(e) \ge 0$ for every $e \in \mathcal {Z}$, and the second inequality follows by Theorem 2.1. $\square$

Corollary 2.7

Let $\{G_1,\dots ,G_r\}$ be a decomposition of the graph G with minimum degree $\delta$. If $d_{v}=\delta$ for every $v\in \mathcal {W}$, then

$$\begin{aligned} GA _1(G) \ge \sum _{i=1}^r GA _1(G_i) - card\, \mathcal {Z} . \end{aligned}$$

Furthermore, if every edge in $\mathcal {Z}$ is maximal, then

$$\begin{aligned} GA _1(G) \ge \sum _{i=1}^r GA _1(G_i) . \end{aligned}$$

The obtention of bounds for the geometric arithmetic index is a very active topic of research (see e.g., [5, 13, 17, 21, 24], the survey [4] and the references therein). The following result provides an upper bound for it involving the numbers of vertices and the diameters of the primary subgraphs in the biconnected decomposition.

Let us define M(n, r) as follows: $M(n,r) := r + \dfrac{1}{2} (n-r-1)(n-r+4)$.

We denote by ${{\mathrm{diam}}}(G)$ the diameter of G, i.e., ${{\mathrm{diam}}}(G):=\max \{d(u,v)\,|\, u,v\in V(G)\} .$

Theorem 2.8

If $\{G_1,\dots ,G_k\}$ is the biconnected decomposition of a graph G and $G_j$ has $n_j$ vertices for $1\le j \le k$, then

$$\begin{aligned} GA _1(G) \le \sum _{j=1}^k M\big (n_j,{{\mathrm{diam}}}(G_j)\big ). \end{aligned}$$

Proof

If $G_j$ has $n_j$ vertices and $m_j$ edges for $1\le j \le k$, then G has $m= \sum _{j=1}^k m_j$ edges. Since each $G_j$ is two-connected, the classical Ore’s result in [14, Theorem 3.1] gives $m_j \le \sum _{j=1}^k M\big (n_j,{{\mathrm{diam}}}(G_j)\big )$ for every $1\le j \le k$. Since $GA _1(G)$ is bounded by the number of edges in G, we have

$$\begin{aligned} GA _1(G) \le m = \sum _{j=1}^k m_j \le \sum _{j=1}^k M\big (n_j,{{\mathrm{diam}}}(G_j)\big ). \end{aligned}$$

$\square$

3 Applications to mathematical chemistry

Next, we apply our results on decompositions in order to compute the geometric-arithmetic indices of several chemical graphs.

Let $\{G_i\}_{i=1}^d$ be a set of finite pairwise disjoint graphs and $v_i,w_i\in V (G_i)$. The chain graph

$$\begin{aligned} C(G_1,G_2, \dots ,G_d) = C(G_1,G_2, \dots ,G_d; v_1,w_1, v_2,w_2, \dots , v_d,w_d) \end{aligned}$$

of $\{G_i\}_{i=1}^d$ with respect to the vertices $\{v_i,w_i\}_{i=1}^d$ is the graph obtained from the graphs $G_1, \dots ,G_d$ by identifying the vertex $w_i$ and the vertex $v_{i+1}$ for every $i \in \{1, 2, \dots , d-1\}$.

We denote by $u_i$ the vertex in $C(G_1,G_2, \dots ,G_d; v_1,w_1,\dots , v_d,w_d)$ obtained by identifying the vertices $w_i$ and $v_{i+1}$. It is clear that $\{G_1, \dots ,G_d\}$ is a T-decomposition of the chain graph:

$$\begin{aligned} C(G_1, \dots ,G_d; v_1,w_1,\dots , v_d,w_d). \end{aligned}$$

Following the notation in [9], given any graph H, $v,w\in V(H)$ and an integer $d>1$, let us define

$$\begin{aligned} T_d(H, v,w) = C(H,\dots ,H; v,w, \dots , v,w), \end{aligned}$$

where H appears d times. Theorem 2.1 has the following direct consequence.

Corollary 3.1

Consider any graph H, $v,w\in V(H)$ and an integer $d>1$.

(1) If $vw \notin E(H)$, then

$$\begin{aligned} GA _1\big ( T_d(H, v,w) \big ) = d\, GA _1(H) + (d-1) W(u_1) . \end{aligned}$$

(2) If $vw \in E(H)$, then

$$\begin{aligned} GA _1\big ( T_d(H, v,w) \big ) = d\, GA _1(H) + (d-1) W(u_1) + (d-2) Z(vw) . \end{aligned}$$

Example 3.2

Spiro chain of hexagons. Consider the chain graph $T_d(C_6)$, where $C_6$ is the cycle graph with vertices $v_1, v_2, \dots , v_6$ (labeled clockwise), $v = v_1$ and $w = v_4$. Since $vw \notin E(C_6)$, Corollary 3.1 gives

$$\begin{aligned} \begin{aligned} GA _1\big ( T_d(C_6) \big )&= d\, GA _1(C_6) + (d-1) W(u_1) \\&= 6d + (d-1) \left( 4\,\frac{2\sqrt{8}}{6}-4\right) = \left( 2+ \frac{8\sqrt{2}}{3} \,\right) d + 4- \frac{8\sqrt{2}}{3}. \end{aligned} \end{aligned}$$

Note that if we take $v = v_3$ or $w = v_5$ in Example 3.2, then we obtain graphs with the same geometric-arithmetic index than the previous one.

Given any graph H, $v,w\in V(H)$, $P_2$ a graph of one edge connecting the two vertices $v',w',$ and an integer $d>1$, let us define

$$\begin{aligned} U_d(H) = C(H,P_2,\dots ,H,P_2,H; v,w,v',w', \dots , v,w,v',w',v,w), \end{aligned}$$

where H appears d times. Theorem 2.1 also has the following consequence.

Corollary 3.3

Consider any graph H, $v,w\in V(H)$ and an integer $d>1$. Denote by $d_v,d_w$ the degrees of v, w in H, respectively.

(1) If $vw \notin E(H)$, then

$$\begin{aligned} GA _1\big ( U_d(H) \big ) = d\, GA _1(H) + (d-1) \left( W(u_1) + W(u_2) + \frac{2\sqrt{(d_{v}+1)(d_{w}+1)}}{d_{v} + d_{w} +2}\,\right) . \end{aligned}$$

(2) If $vw \in E(H)$, then

$$\begin{aligned} GA _1\big ( U_d(H) \big )= & {} d\, GA _1(H) + (d-1) \left( W(u_1) + W(u_2) + \frac{2\sqrt{(d_{v}+1)(d_{w}+1)}}{d_{v} + d_{w} +2}\,\right) \\&+ (d-2)Z(vw) . \end{aligned}$$

Proof

Consider the T-decomposition $\{H,P_2,\dots ,H,P_2,H\}$ of $U_d(H)$. Note that $GA _1(P_2)=1$ and $Z(P_2)= \frac{2\sqrt{(d_{v}+1)(d_{w}+1)}}{d_{v} + d_{w} +2}-1$. If $vw \notin E(H)$, then $\mathcal {Z}$ is just the set of copies of $P_2$. Hence, Theorem 2.1 gives

$$\begin{aligned} \begin{aligned} GA _1\big ( U_d(H) \big )&= d\, GA _1(H) + d-1 + (d-1) W(u_1) + (d-1) W(u_2) \\&\quad + (d-1)\left( \frac{2\sqrt{(d_{v}+1)(d_{w}+1)}}{d_{v} + d_{w} +2} - 1 \right) \\&= d\, GA _1(H) + (d-1) \left( W(u_1) + W(u_2) + \frac{2\sqrt{(d_{v}+1)(d_{w}+1)}}{d_{v} + d_{w} +2}\,\right) . \end{aligned} \end{aligned}$$

If $vw \in E(H)$, then $\mathcal {Z}$ is the set of copies of $P_2$ and the copies of vw which are not in the first and last copies of H. Therefore, we just need to add the term $(d-2)Z(vw)$ to the formula in (1). $\square$

Example 3.4

Polyphenylenes. Consider the chain graph $U_d(C_6)$, where $C_6$ is the cycle graph with vertices $v_1, v_2, \dots , v_6$ (labeled clockwise), $v = v_1$ and $w = v_4$. Since $vw \notin E(C_6)$, Corollary 3.3 gives

$$\begin{aligned} \begin{aligned} GA _1\big ( U_d(C_6) \big )&= d\, GA _1(C_6) + (d-1) \left( W(u_1) + W(u_2) + \frac{2\sqrt{(d_{v}+1)(d_{w}+1)}}{d_{v} + d_{w} +2}\,\right) \\&= 6d \!+ \!(d\!-\!1) \left( 2\left( 2\frac{2\sqrt{6}}{5}\!-\!2\right) \!+\! 1\right) \!=\! \left( 3\!+\! \frac{8\sqrt{6}}{5} \,\right) d + 3- \frac{8\sqrt{6}}{5}. \end{aligned} \end{aligned}$$

If we take $v = v_3$ or $w = v_5$ in Example 3.4, then we obtain graphs with the same geometric-arithmetic index than the previous one.

Let $\{G_i\}_{i=1}^d$ be a set of finite pairwise disjoint graphs and $v_i\in V (G_i)$. The bridge graph

$$\begin{aligned} B(G_1,G_2, \dots ,G_d) = B(G_1,G_2, \dots ,G_d; v_1, v_2, \dots , v_d) \end{aligned}$$

of $\{G_i\}_{i=1}^d$ with respect to the vertices $\{v_i\}_{i=1}^d$ is the graph obtained from the graphs $G_1, \dots ,G_d$ by connecting the vertices $v_i$ and $v_{i+1}$ by an edge for every $i \in \{1, 2, \dots , d-1\}$. Theorem 2.1 has the following consequence.

Theorem 3.5

Let $\{G_i\}_{i=1}^d$ be a set of finite pairwise disjoint graphs and $v_i\in V (G_i)$. Then

$$\begin{aligned} GA _1\big (B(G_1, \dots ,G_d; v_1, \dots , v_d)\big )= & {} \sum _{i=1}^d GA _1(G_i) + d-1 + \sum _{i=1}^d W(v_i) \\&+ \sum _{i=1}^{d-1} Z(v_i v_{i+1}). \end{aligned}$$

Proof

Consider the T-decomposition $\{G_1, \dots , G_d, v_1v_2, \dots , v_{d-1}v_{d}\}$ of

$$\begin{aligned} B(G_1, \dots ,G_d; v_1, \dots , v_d). \end{aligned}$$

We have in this case $\mathcal {W}=\{v_1, \dots , v_d\}$, $\mathcal {Z}=\{v_1v_2, \dots , v_{d-1}v_d\}$ and $GA _1(v_i v_{i+1})=1$ for $i\in \{1,\dots ,d-1\}$. Hence, Theorem 2.1 gives the result. $\square$

Given any graph H, $v\in V(H)$ and an integer $d>1$, let us define

$$\begin{aligned} G_d(H,v) = B(H,\dots ,H; v, \dots , v), \end{aligned}$$

where H appears d times. Theorem 3.5 has the following corollary.

Corollary 3.6

Consider a graph H, $v\in V(H)$ and an integer $d>1$. Denote by $v_i$ the copy of v in the i-th copy of H in $G_d(H,v)$ and by $d_v$ the degree of v in H.

(1) If $d=2$, then

$$\begin{aligned} GA _1\big ( G_d(H,v) \big ) = d\, GA _1(H) +1 + 2 W(v_1) . \end{aligned}$$

(2) If $d \ge 3$, then

$$\begin{aligned} GA _1\big ( G_d(H,v) \big )= & {} d\, GA _1(H) + d-3 + 2 W(v_1) + (d - 2) W(v_2) \\&+ \frac{4\sqrt{(d_{v}+1) (d_{v}+2)}}{2d_{v}+3}. \end{aligned}$$

Proof

We prove first the case (2). Note that $Z(v_{1}v_2) = Z(v_{d-1}v_d)= \frac{2\sqrt{(d_{v}+1) (d_{v}+2)}}{2d_{v}+3} -1$ and $Z(v_i v_{i+1})=1-1=0$ for every $i\in \{2,3,\dots ,d-2\}$. Theorem 3.5 gives

$$\begin{aligned} GA _1\big (G_d(H,v)\big )= & {} d\, GA _1(H) + d-1 + \sum _{i=1}^d W(v_i) + \sum _{i=1}^{d-1} Z(v_i v_{i+1}) \\= & {} d\, GA _1(H) + d-1 + 2 W(v_1) + (d- 2) W(v_2) \\&+\, Z(v_{1}v_2) + Z(v_{d-1}v_d) \\= & {} d\, GA _1(H) + d-1 + 2 W(v_1) + (d - 2) W(v_2) \\&+\, \frac{4\sqrt{(d_{v}+1) (d_{v}+2)}}{2d_{v}+3} -2. \end{aligned}$$

In case (1) we have $\mathcal {Z}=\{v_1v_2\}$ and $Z(v_1 v_{1})=1-1=0$, and so the previous argument gives the formula. $\square$

Example 3.7

As an application of the previous result, we consider $k\ge 3$, $d \ge 3$ and the bridge graph $G_d(P_k,w_1)$, where $P_k$ is the path graph with vertices $w_1, w_2, \dots , w_k$. Corollary 3.6 gives

$$\begin{aligned} GA _1\big ( G_d(P_k,w_1) \big )= & {} d\, GA _1(P_k) + d-3 + 2 W(v_1) + (d - 2) W(v_2) \\&+\, \frac{4\sqrt{(d_{v}+1) (d_{v}+2)}}{2d_{v}+3} \\= & {} \left( k-3+\frac{4\sqrt{2}}{3}\,\right) d + d-3 + 2 \left( 1-\frac{2\sqrt{2}}{3}\, \right) \\&+\, (d- 2) \left( \frac{2\sqrt{3}}{4} -\frac{2\sqrt{2}}{3} \, \right) + \frac{4\sqrt{6}}{5} \\= & {} \left( k-2+\frac{2\sqrt{2}}{3}+ \frac{\sqrt{3}}{2} \,\right) d - 1 -\sqrt{3} + \frac{4\sqrt{6}}{5}. \end{aligned}$$

Example 3.8

Polyethene. For $d \ge 3$, consider the bridge graph $G_d(P_3,w_2)$. Note that $G_4(P_3,w_2)$ is the graph corresponding to Polyethene. Corollary 3.6 gives

$$\begin{aligned} GA _1\big ( G_d(P_3,w_2) \big )= & {} d\, GA _1(P_3) + d-3 + 2 W(v_1) + (d- 2) W(v_2) \\&+\, \frac{4\sqrt{(d_{v}+1) (d_{v}+2)}}{2d_{v}+3} \\= & {} \frac{4\sqrt{2}}{3}\, d + d-3+ 2 \left( 2 \left( \frac{2\sqrt{3}}{4} - \frac{2\sqrt{2}}{3}\, \right) \right) \\&+\,(d- 2) \left( 2 \left( \frac{2\sqrt{4}}{5} - \frac{2\sqrt{2}}{3}\, \right) \right) + \frac{4\sqrt{12}}{7} \\= & {} \frac{13}{5} \, d - \frac{31}{5} + \frac{22\sqrt{3}}{7}. \end{aligned}$$

Note that in the previous results we have used always T-decompositions. We deal now with a class of graphs obtained by decompositions which are not T-decompositions.

Let $\{G_i\}_{i=1}^d$ ($d\ge 3$) be a set of finite pairwise disjoint graphs and $v_i\in V (G_i)$. The circle graph $F(G_1,G_2, \dots ,G_d) = F(G_1,G_2, \dots ,G_d; v_1, v_2, \dots , v_d)$ of $\{G_i\}_{i=1}^d$ with respect to the vertices $\{v_i\}_{i=1}^d$ is the graph obtained from the graphs $G_1, \dots ,G_d$ by connecting the vertices $v_i$ and $v_{i+1}$ by an edge for every $i \in \{1, 2, \dots , d\}$, where $v_{d+1}$ is defined as $v_{d+1}=v_1$. Theorem 2.1 has the following consequence.

Theorem 3.9

Let $\{G_i\}_{i=1}^d$ be a set of finite pairwise disjoint graphs with $d\ge 3$ and $v_i\in V (G_i)$. Then

$$\begin{aligned} GA _1\big (F(G_1, \dots ,G_d; v_1, \dots , v_d)\big )= & {} \sum _{i=1}^d GA _1(G_i) + d + \sum _{i=1}^d W(v_i) \\&\quad + \sum _{i=1}^{d} Z(v_i v_{i+1}). \end{aligned}$$

Proof

Consider the decomposition $\{G_1, \dots , G_d, v_1v_2, \dots , v_{d}v_{d+1}\}$ of

$$\begin{aligned} F(G_1, \dots ,G_d; v_1, \dots , v_d). \end{aligned}$$

Since we have $\mathcal {W}=\{v_1, \dots , v_d\}$, $\mathcal {Z}=\{v_1v_2, \dots , v_{d}v_{d+1}\}$ and $GA _1(v_i v_{i+1})=1$ for $i\in \{1,\dots ,d\}$, Theorem 2.1 gives the result. $\square$

Given any graph H, $v\in V(H)$ and an integer $d>1$, let us define $F_d(H,v) = F(H,\dots ,H; v, \dots , v),$ where H appears d times. Theorem 3.9 has the following corollary.

Corollary 3.10

Consider a graph H, $v\in V(H)$ and an integer $d\ge 3$. Denote by $v_1$ the copy of v in the first copy of H in $F_d(H,v)$. Then

$$\begin{aligned} GA _1\big ( F_d(H,v) \big ) = d\, GA _1(H) + d+ d\,W(v_1). \end{aligned}$$

Proof

Note that $Z(v_i v_{i+1})=1-1=0$ for every $i\in \{1,\dots ,d\}$. Theorem 3.9 gives

$$\begin{aligned} \begin{aligned} GA _1\big (F_d(H,v)\big )&= d\, GA _1(H) + d + \sum _{i=1}^d W(v_i) + \sum _{i=1}^{d} Z(v_i v_{i+1}) \\&= d\, GA _1(H) + d + d\,W(v_1). \end{aligned} \end{aligned}$$

$\square$

Let $\{G_i\}_{i=1}^d$ be a set of finite pairwise disjoint graphs and $v_i\in V (G_i)$. The bouquet $S(G_1,G_2, \dots ,G_d) = S(G_1,G_2, \dots ,G_d; v_1, v_2, \dots , v_d)$ of $\{G_i\}_{i=1}^d$ with respect to the vertices $\{v_i\}_{i=1}^d$ is the graph obtained from the graphs $G_1, \dots ,G_d$ by identifying the vertices $\{v_i\}_{i=1}^d$ with a new vertex v. Theorem 2.1 has the following consequence.

Theorem 3.11

Let $\{G_i\}_{i=1}^d$ be a set of finite pairwise disjoint graphs and $v_i\in V (G_i)$. Then $GA _1\big (S(G_1, \dots ,G_d; v_1, \dots , v_d)\big ) = \sum _{i=1}^d GA _1(G_i) + W(v) .$

Proof

Consider the T-decomposition $\{G_1, \dots , G_d\}$ of $S(G_1, \dots ,G_d; v_1, \dots , v_d)$. Note that we have in this case $\mathcal {W}=\{v\}$ and so $\mathcal {Z}=\emptyset$. Hence, Theorem 2.1 gives the result. $\square$

Given any graph H, $v\in V(H)$ and an integer $d>1$, let us define

$$\begin{aligned} S_d(H,v) = S(H,\dots ,H; v, \dots , v), \end{aligned}$$

where H appears d times. Theorem 3.11 has the following corollary.

Corollary 3.12

Consider a graph H, $v\in V(H)$ and an integer $d>1$. Denote by $d_v$ the degree of v in H. Then

$$\begin{aligned} GA _1\big ( S_d(H,v) \big ) = d\,\left( GA _1(H) + \sum _{u\in N_{H}(v)}\frac{2\sqrt{d\, d_u d_v}}{ d_u + d\,d_v} - \sum _{ u\in N_{H}(v)} \frac{2\sqrt{d_u d_{v}}}{d_u + d_{v}}\,\right) . \end{aligned}$$

Proof

Since $\mathcal {W}=\{v\}$, we have $N_H(v) {\setminus } \mathcal {W} = N_H(v)$. Each copy of H contributes to W(v) with

$$\begin{aligned} \sum _{u\in N_{H}(v)}\frac{2\sqrt{d\, d_u d_v}}{ d_u + d\,d_v} - \sum _{ u\in N_{H}(v)} \frac{2\sqrt{d_u d_{v}}}{d_u + d_{v}}\,, \end{aligned}$$

and so Theorem 3.11 gives the result. $\square$