New bounds for Randic and GA indices

Abstract

The main goal of this paper is to present some new lower and upper bounds for the Randic and GA indices in terms of Zagreb and modified Zagreb indices.

MSC: 05C05, 05C20, 05C90.

Dedication

Dedicated to Professor Hari M Srivastava.

1 Introduction and preliminaries

A systematic study of topological indices is one of the most striking aspects in many branches of mathematics with its applications and various other fields of science and technology. A topological index is a numeric quantity from the structural graph of a molecule. Usage of topological indices in chemistry began in 1947 when H. Wiener developed the most widely known topological descriptor, namely the Wiener index, and used it to determine physical properties of types of alkanes known as paraffin (see, for instance, [1–3]).

Let G be a simple graph with the vertex-set $V(G)$ and the edge-set $E(G)$. As usual notion, the maximum vertex degree is denoted by $\mathrm{\Delta }=\mathrm{\Delta }(G)$, while the minimum vertex degree is denoted by $\delta =\delta (G)$. Moreover, ${\delta }_1={\delta }_1(G)$ denotes the minimum nonpendant vertex degree in G. A vertex of the graph G is said to be pendant if its neighborhood contains exactly one vertex. On the other hand, an edge of a graph is said to be pendant if one of its vertices is pendant.

In 1975, Randic [4] introduced the connectivity index, namely Randic index, to reflect molecular branching. In fact, the Randic index is defined as

$$\chi (G)=\sum _{uv\in E(G)}\frac{1}{\sqrt{d_u{d}_v}}.$$

(1)

Furthermore, again by considering the degrees of vertices in G, Vukicević and Furtula [5] developed the Geometric-arithmetic index, shortly GA index, which is defined by

$$GA(G)=\sum _{uv\in E(G)}\frac{2\sqrt{d_u{d}_v}}{d_u+d_v}.$$

(2)

In the following, we recall two fundamental indices that will be used to present some new bounds for Randic and GA indices.

The (first and second) Zagreb indices have been introduced by Gutman and Trinajstić [6] as the form

$$M_1(G)=\sum _{v\in V(G)}{(d_v)}^2\text{and}{M}_2(G)=\sum _{uv\in E(G)}{d}_u{d}_v,$$

(3)

where $d_u$ and $d_v$ are the degrees of u and v, respectively. On the other hand, for a (molecular) graph G, the modified second Zagreb index $M_2^{\ast }(G)$ is defined as

$$M_2^{\ast }(G)=\sum _{uv\in E(G)}\frac{1}{d_u{d}_v}$$

(4)

(cf. [7–10]).

This paper is organized as follows. In the forthcoming section, we present lower and upper bounds on Randic index of connected graphs and trees in terms of modified Zagreb indices given in (4). The final section deals with lower and upper bounds on GA index of connected graphs and trees in terms of Zagreb indices given in (3). We note that this paper is motivated from [11].

2 Lower and upper bounds on Randic index

Throughout this paper, we refer the book [12] for a classical result, namely the Pólya-Szegó inequality. From this result, we first establish the following theorem, which will be expressed the lower bound on the Randic index.

Theorem 1 Let G be a simple connected graph of order n with m edges, and let p, Δ and ${\delta }_1$ denote the number of pendant vertices, maximum vertex degree and minimum nonpendant vertex degree of G, respectively. Then

$$\chi (G)\ge \frac{p}{\sqrt{\mathrm{\Delta }}}+\frac{2\sqrt{{\delta }_1\mathrm{\Delta }(m-p)}}{{\delta }_1+\mathrm{\Delta }}\sqrt{M_2^{\ast }(G)-\frac{p}{\mathrm{\Delta }}}.$$

Proof For $2\le {\delta }_1\le d_i,d_j\le \mathrm{\Delta }$, we clearly have

$$\frac{1}{d_i{d}_j}\ge \frac{1}{d_i\mathrm{\Delta }}\ge \frac{1}{{\mathrm{\Delta }}^2}$$

such that the equality holds if and only if $d_i=d_j=\mathrm{\Delta }$. We also have

$$\frac{1}{d_i{d}_j}\le \frac{1}{d_i{\delta }_1}\le \frac{1}{{\delta }_1^2}$$

with equality holding if and only if $d_i=d_j={\delta }_1$.

Since p is the number of pendant vertices in G, we have total $m-p$ number of non-pendant edges in G. By the Pólya-Szegó inequality, we have

$$\begin{array}{rcl}{\left(\sum _{v_i{v}_j\in E(G):d_j,d_j\ne 1}\frac{1}{\sqrt{d_i{d}_j}}\right)}^2& \ge & \frac{4{\delta }_1\mathrm{\Delta }(m-p)}{{({\delta }_1+\mathrm{\Delta })}^2}\left(\sum _{v_i{v}_j\in E(G):d_i,d_j\ne 1}\frac{1}{d_i{d}_j}\right)\\ \ge & \frac{4{\delta }_1\mathrm{\Delta }(m-p)}{{({\delta }_1+\mathrm{\Delta })}^2}(M_2^{\ast }(G)-\sum _{v_i{v}_j\in E(G):d_i=1}\frac{1}{d_j}).\end{array}$$

This inequality can be clearly written as

$$\sum _{v_i{v}_j\in E(G):d_j,d_j\ne 1}\frac{1}{\sqrt{d_i{d}_j}}\ge \frac{\sqrt{4{\delta }_1\mathrm{\Delta }(m-p)}}{({\delta }_1+\mathrm{\Delta })}\sqrt{M_2^{\ast }(G)-p\frac{1}{\mathrm{\Delta }}}.$$

(5)

From (1), we get

$$\chi (G)=\sum _{v_i{v}_j\in E(G):d_i=1}\frac{1}{\sqrt{d_j}}+\sum _{v_i{v}_j\in E(G):d_i,d_j\ne 1}\frac{1}{\sqrt{d_i{d}_j}}.$$

(6)

For $\mathrm{\Delta }\ge d_i$, since $\frac{1}{d_i}\ge \frac{1}{\mathrm{\Delta }}$, by (5) and (6), we obtain

$$\chi (G)\ge \frac{p}{\sqrt{\mathrm{\Delta }}}+\frac{2\sqrt{{\delta }_1\mathrm{\Delta }(m-p)}}{{\delta }_1+\mathrm{\Delta }}\sqrt{M_2^{\ast }(G)-\frac{p}{\mathrm{\Delta }}},$$

as desired. □

Corollary 1 Let T be a tree of order n with p pendant vertices, and let Δ and ${\delta }_1$ be the maximum vertex and minimum nonpendent vertex degrees of T, respectively. Then

$$\chi (T)\ge \frac{p}{\sqrt{\mathrm{\Delta }}}+\frac{2\sqrt{{\delta }_1\mathrm{\Delta }(n-1-p)}}{{\delta }_1+\mathrm{\Delta }}\sqrt{M_2^{\ast }(G)-\frac{p}{\mathrm{\Delta }}}.$$

Proof Since the number of edges in a tree having n vertices is $m=n-1$, the proof can be done similarly as in the proof of Theorem 1. □

Theorem 2 Let G be a simple connected graph of order n with m edges, and let p, Δ and ${\delta }_1$ denote the number of pendant vertices, maximum vertex degree and minimum nonpendant vertex degree of G, respectively. Then

$$\chi (G)\le \frac{p}{\sqrt{{\delta }_1}}+\sqrt{(m-p)(M_2^{\ast }(G)-\frac{p}{{\delta }_1})}.$$

Proof By the Cauchy-Schwarz inequality, it is clear that

$$\begin{array}{rcl}{\left(\sum _{v_i{v}_j\in E(G):d_j,d_j\ne 1}\frac{1}{\sqrt{d_i{d}_j}}\right)}^2& \le & (m-p)\left(\sum _{v_i{v}_j\in E(G):d_j,d_j\ne 1}\frac{1}{d_i{d}_j}\right)\\ \le & (m-p)(M_2^{\ast }(G)-\sum _{v_i{v}_j\in E(G):d_i=1}\frac{1}{d_j})\\ \le & (m-p)(M_2^{\ast }(G)-\frac{p}{{\delta }_1})\end{array}$$

which can be rewritten as

$$\sum _{v_i{v}_j\in E(G):d_j,d_j\ne 1}\frac{1}{\sqrt{d_i{d}_j}}\le \sqrt{(m-p)(M_2^{\ast }(G)-\frac{p}{{\delta }_1})}.$$

(7)

Since $\frac{1}{d_j}\le \frac{1}{{\delta }_1}$ for ${\delta }_1\le d_j$, by (5) and (7), we obtain

$$\chi (G)\le \frac{p}{\sqrt{{\delta }_1}}+\sqrt{(m-p)(M_2^{\ast }(G)-\frac{p}{{\delta }_1})},$$

as required. □

Now we prove another form of the upper bound for the Randic index as in the following.

Theorem 3 Let G be a simple connected graph of order n with m edges, and let p, Δ and ${\delta }_1$ denote the number of pendant vertices, maximum vertex degree and minimum nonpendant vertex degree of G, respectively. Then

$$\chi (G)\le \frac{p}{\sqrt{{\delta }_1}}+\frac{(m-p)}{{\delta }_1}.$$

(8)

Proof Since $\frac{1}{{\delta }_1^2}$ is the maximum value of $\frac{1}{d_i{d}_j}$ for all edges $v_i{v}_j\in E(G)$, we have

$$\begin{array}{rcl}{M}_2^{\ast }(G)-\sum _{v_i{v}_j\in E(G):d_i=1}\frac{1}{d_j}& =& \sum _{v_i{v}_j\in E(G):d_j,d_j\ne 1}\frac{1}{\sqrt{d_i{d}_j}}\\ \le & \frac{m-p}{{\delta }_1^2}.\end{array}$$

(9)

After that, by using (9) in (5), we get the bound in (8), as required. □

3 Lower and upper bounds on GA index

By taking Pólya-Szegó inequality into account, the next result deals with a new lower bound on GA index in terms of Zagreb index as given in (3).

Theorem 4 Let G be a simple connected graph of order n with m edges, and let p, Δ and ${\delta }_1$ denote the number of pendant vertices, maximum vertex degree and minimum nonpendant vertex degree of G, respectively. Then

$$GA(G)\ge \frac{2p\sqrt{{\delta }_1}}{1+\mathrm{\Delta }}+2\sqrt{2}\frac{{\delta }_1\mathrm{\Delta }}{({\delta }_1^2+{\mathrm{\Delta }}^2)}\sqrt{\frac{(m-p)}{\mathrm{\Delta }}(M_2(G)-p{\delta }_1)}.$$

Proof For $2\le {\delta }_1\le d_i,d_j\le \mathrm{\Delta }$, we have

$$\frac{1}{2\mathrm{\Delta }}\le \frac{1}{(d_i+d_j)}\le \frac{1}{2{\delta }_1}$$

which implies

$$\frac{d_i{d}_j}{{(d_i+d_j)}^2}\le \frac{{\mathrm{\Delta }}^2}{4{\delta }_1^2}.$$

On the other hand, since we also have

$$\frac{d_i{d}_j}{{(d_i+d_j)}^2}\ge \frac{{\delta }_1^2}{4{\mathrm{\Delta }}^2},$$

the combination of these above equalities implies that

$$\frac{{\delta }_1}{\mathrm{\Delta }}\le \frac{2\sqrt{d_i{d}_j}}{(d_i+d_j)}\le \frac{\mathrm{\Delta }}{{\delta }_1}.$$

(10)

Since p is the number of pendant vertices in G, we have total $m-p$ number of non-pendant edges in G. By the Pólya-Szegó inequality, we get

$$\begin{array}{rcl}{\left(\sum _{v_i{v}_j\in E(G):d_j,d_j\ne 1}\frac{2\sqrt{d_i{d}_j}}{(d_i+d_j)}\right)}^2& \ge & \frac{4{\delta }_1^2{\mathrm{\Delta }}^2(m-p)}{{({\delta }_1^2+{\mathrm{\Delta }}^2)}^2}\left(\sum _{v_i{v}_j\in E(G):d_i,d_j\ne 1}\frac{4d_i{d}_j}{(d_i+d_j)}\right)\\ \ge & \frac{4{\delta }_1^2{\mathrm{\Delta }}^2(m-p)}{{({\delta }_1^2+{\mathrm{\Delta }}^2)}^2}\left(\sum _{v_i{v}_j\in E(G):d_i,d_j\ne 1}\frac{4d_i{d}_j}{2\mathrm{\Delta }}\right)\\ \ge & \frac{8{\delta }_1^2{\mathrm{\Delta }}^2(m-p)}{\mathrm{\Delta }{({\delta }_1^2+{\mathrm{\Delta }}^2)}^2}(M_2(G)-\sum _{v_i{v}_j\in E(G):d_i=1}{d}_j)\\ \ge & \frac{8{\delta }_1^2{\mathrm{\Delta }}^2(m-p)}{\mathrm{\Delta }{({\delta }_1^2+{\mathrm{\Delta }}^2)}^2}(M_2(G)-p{\delta }_1).\end{array}$$

This calculation can be rewritten basically as follows:

$$\sum _{v_i{v}_j\in E(G):d_j,d_j\ne 1}\frac{2\sqrt{d_i{d}_j}}{(d_i+d_j)}\ge 2\sqrt{2}\frac{{\delta }_1\mathrm{\Delta }}{({\delta }_1^2+{\mathrm{\Delta }}^2)}\sqrt{\frac{(m-p)}{\mathrm{\Delta }}(M_2(G)-p{\delta }_1)}.$$

From (2), we obtain

$$GA(G)=\sum _{v_i{v}_j\in E(G):d_i=1}\frac{2\sqrt{d_j}}{(1+d_j)}+\sum _{v_i{v}_j\in E(G):d_i,d_j\ne 1}\frac{2\sqrt{d_i{d}_j}}{(d_i+d_j)}.$$

(11)

Now, for ${\delta }_1\le d_j\le \mathrm{\Delta }$, since $\sqrt{d_j}\ge \sqrt{{\delta }_1}$ and $\frac{1}{1+d_j}\ge \frac{1}{1+\mathrm{\Delta }}$, by (10) and (11), we arrive at

$$GA(G)\ge \frac{2p\sqrt{{\delta }_1}}{1+\mathrm{\Delta }}+2\sqrt{2}\frac{{\delta }_1\mathrm{\Delta }}{({\delta }_1^2+{\mathrm{\Delta }}^2)}\sqrt{\frac{(m-p)}{\mathrm{\Delta }}(M_2(G)-p{\delta }_1)}.$$

Hence the result. □

Corollary 2 Let T be a tree of order n with p pendant vertices, and let Δ and ${\delta }_1$ denote the maximum vertex degree and minimum non-pendent vertex degree of T, respectively. Then

$$GA(G)\ge \frac{2p\sqrt{{\delta }_1}}{1+\mathrm{\Delta }}+2\sqrt{2}\frac{{\delta }_1\mathrm{\Delta }}{({\delta }_1^2+{\mathrm{\Delta }}^2)}\sqrt{\frac{(n-1-p)}{\mathrm{\Delta }}(M_2(G)-p{\delta }_1)}.$$

Proof For an order n, since the number of edges in a tree T is $m=n-1$, the proof can be done quite similar as the proof of Theorem 4. □

Theorem 5 Let G be a simple connected graph of order n with m edges, and let p, Δ and ${\delta }_1$ denote the number of pendant vertices, maximum vertex degree and minimum non-pendant vertex degree of G, respectively. Then

$$GA(G)\le \frac{2p\sqrt{\mathrm{\Delta }}}{1+{\delta }_1}+\frac{1}{{\delta }_1}\sqrt{(m-p)(M_2(G)-p\mathrm{\Delta })}.$$

Proof By the Cauchy-Schwarz inequality,

$$\begin{array}{rcl}{\left(\sum _{v_i{v}_j\in E(G):d_j,d_j\ne 1}\frac{2\sqrt{d_i{d}_j}}{(d_i+d_j)}\right)}^2& \le & (m-p)\left(\sum _{v_i{v}_j\in E(G):d_j,d_j\ne 1}\frac{4d_i{d}_j}{{(d_i+d_j)}^2}\right)\\ \le & (m-p)\left(\sum _{v_i{v}_j\in E(G):d_j,d_j\ne 1}\frac{d_i{d}_j}{{\delta }_1^2}\right)\\ \le & \frac{(m-p)}{{\delta }_1^2}(M_2(G)-\sum _{v_i{v}_j\in E(G):d_i=1}{d}_j)\\ \le & \frac{(m-p)}{{\delta }_1^2}(M_2(G)-p\mathrm{\Delta })\end{array}$$

which can be simply indicate as

$$\sum _{v_i{v}_j\in E(G):d_j,d_j\ne 1}\frac{2\sqrt{d_i{d}_j}}{(d_i+d_j)}\le \frac{1}{{\delta }_1}\sqrt{(m-p)(M_2(G)-p\mathrm{\Delta })}.$$

(12)

Now, for ${\delta }_1\le d_j\le \mathrm{\Delta }$, since $\sqrt{d_j}\le \sqrt{\mathrm{\Delta }}$ and $\frac{1}{1+d_j}\le \frac{1}{1+{\delta }_1}$, by (11) and (12) we get the result, as required. □

The following theorem presents another upper bound for GA index.

Theorem 6 Let G be a simple connected graph of order n with m edges, and let p, Δ and ${\delta }_1$ denote the number of pendant vertices, maximum vertex degree and minimum non-pendant vertex degree of G, respectively. Then

$$G(A)\le \frac{2p\sqrt{\mathrm{\Delta }}}{1+{\delta }_1}+\frac{(m-p)\mathrm{\Delta }}{{\delta }_1}.$$

Proof Since ${\mathrm{\Delta }}^2$ is the maximum value of $d_i{d}_j$ for all edges $v_i{v}_j\in E(G)$, we have

$$\begin{array}{rcl}{M}_2(G)-\sum _{v_i{v}_j\in E(G):d_i=1}{d}_j& =& \sum _{v_i{v}_j\in E(G):d_j,d_j\ne 1}{d}_i{d}_j\\ \le & (m-p){\mathrm{\Delta }}^2.\end{array}$$

(13)

Now, by using (13) in (11), we get

$$G(A)\le \frac{2p\sqrt{\mathrm{\Delta }}}{1+{\delta }_1}+\frac{(m-p)\mathrm{\Delta }}{{\delta }_1}.$$

Hence, the result. □