Relations Between the Energy and Topological Indices of a Graph

Abstract

In this paper, we give various lower and upper bounds for the energy of graphs in terms of several topological indices of graphs: the first general multiplicative Zagreb index, the general Randić index, the general zeroth-order Randić index, the redefined Zagreb indices, and the atom-bond connectivity index. Moreover, we obtain new bounds for the energy in terms of certain graph invariants as diameter, girth, algebraic connectivity and radius.

1 Introduction

Let $G = (V, E)$ be a simple undirected graph and let $n=|V({G})|$ and $m=|E({G})|$ be the order and the size of the graph G, respectively. The open neighborhood of vertex v is the set $N_{G}(v)=\{u\in V({G})\mid uv\in E({G})\}$ and the degree of $v_i$ is defined as $d_{v_i}=|N(v_i)|$. Let $\Delta$ and $\delta$ be the maximum and the minimum degree of G, respectively. A simple undirected graph in which every pair of distinct vertices is connected by a unique edge, we call complete graph and is denoted by $K_n$.

For a vertex v in a connected nontrivial graph G, with $ecc_G(v) = \max \{d_G(v, u)\mid u \in V (G)\}$ we denote the eccentricity of v. The radius r(G) of G is defined as $r(G) = \min \{ecc_G(v)\mid v \in V (G)\}$. The diameter of a graph G is the maximum distance between two vertices of G; denoted by D(G). A girth g(G) of a graph G is the length of the shortest cycle in the graph.

Topological indices represent an important type of molecular descriptors. They have gained considerable popularity and many new topological indices have been proposed and studied in the mathematical chemistry literature in recent years.

Various generalizations of the Zagreb indices have been proposed. In [18] a so-called general zeroth-order Randić index was introduced. It is defined as

$$\begin{aligned} ^{0}R_\alpha (G) =\sum _{v\in V(G)}d_v^{\alpha } \end{aligned}$$

where $\alpha$ is a real number. Note that $^{0}R_{-1}$ is the inverse index ID(G), $^{0}R_{2}(G)$ is the first Zagreb index $M_1(G)$, $^{0}R_{3}(G)$ is the forgotten topological index F(G).

The general Randić index, $R_\alpha (G)$, is a generalization of the second Zagreb index, reported in [5]. This index is defined as

$$\begin{aligned} R_\alpha (G) =\sum _{uv\in E(G)}(d_ud_v)^{\alpha }. \end{aligned}$$

Note that $R_{-1/2}(G)$ is the usual Randić index R(G).

The general redefined first Zagreb index is defined as

$$\begin{aligned} ReZg_\alpha (G) =\sum _{uv\in E(G)}\left( \frac{d_u+d_v}{d_ud_v}\right) ^{\alpha }, \end{aligned}$$

for any real number $\alpha$.

The first general multiplicative Zagreb index of a graph G is defined in [27] as

$$\begin{aligned} P^{\alpha }(G) =\prod _{v\in V(G)}d^{\alpha }_v. \end{aligned}$$

These indices are a generalization of the well-known multiplicative Zagreb indices. If $\alpha = 1$, then $P^1(G)$ is the Narumi-Katayama index NK(G), see [23].

The atom-bond connectivity index of G, denoted by ABC(G), is defined in [9] as

$$\begin{aligned} ABC(G) =\sum _{uv\in E(G)}\sqrt{\frac{d_u+d_v-2}{d_ud_v}}. \end{aligned}$$

The Laplacian matrix L of the graph G is defined as follows: $L_{uv} = 1$ if $(u, v)\in E(G), L_{uv} = 0$ if $(u, v) \notin E(G)$ and $u\ne v$, and $L_{uu} = -d_u$. The algebraic connectivity $\alpha$ of a graph G is the second smallest eigenvalue of the Laplacian matrix L.

The adjacency matrix A(G) of G is defined by its entries as $a_{ij} = 1$ if $(v_i, v_j) \in E(G)$ and 0 otherwise. Let $\lambda _1\geqslant \lambda _2\geqslant \dots \geqslant \lambda _n$ denote the eigenvalues of A(G). The largest eigenvalues of A, $\lambda _1$, is called a spectral radius of the graph G.

The energy of the graph G is defined as

$$\begin{aligned} \mathcal {E}=\mathcal {E}(G)=\sum _{i=1}^{n} |\lambda _i|, \end{aligned}$$

(1)

where $\lambda _i$, $i=1, 2, \dots , n$, are the eigenvalues of the graph G.

This concept was introduced by I. Gutman and is intensively studied in chemistry, since it can be used to approximate the total $\pi$-electron energy of a molecule (see, e.g. [12, 13, 15]). Since then, numerous bounds for the energy were found (see, e.g. [24, 25]).

In the last years, the energy of graphs was related to vertex-degree-based topological indices [3].

Definition 1

The energy of the vertex $v_i$ with respect to G, which is denoted by $\mathcal {E}_G(v_i)$, is given by

$$\begin{aligned} \mathcal {E}_G(v_i) = |A(G)|_{ii} \end{aligned}$$

for $i = 1, \ldots , n$, where $|A| = (AA^{*})^{1/2}$ and A is the adjacency matrix of G.

In this way, the energy of a graph is given by the sum of the individual energies of the vertices of G,

$$\begin{aligned} \mathcal {E}(G) = \sum _{i=1}^{n}\mathcal {E}_G(v_i). \end{aligned}$$

The energy of a vertex should be understood as the contribution of the vertex to the energy of the graph, in terms of how it interacts with other vertices. It can be seen that the energy of a vertex only depends on the vertices that are in the same component as v. Among graph descriptors used in mathematical chemistry, two of them play a rather important role and have the attention of many researchers around the world: the energy and topological indices of a graph. There are many inequalities for each of these descriptors. However, just a few relationship between them have been established, see in [4]. In this paper, we establish new relations between the energy and some of the topological indices of graphs.

For a graph G with Randić index R(G), Arizmendi et al. [4] recently proved that $\mathcal {E}(G) \ge 2R(G)$, where the equality holds if and only if G is the union of complete bipartite graphs. Yan et al. [28] showed that $\mathcal {E}(G) \le 2\sqrt{\Delta }R(G)$. Filipovski [10] obtained relations between the energy of graphs and the Randić index. Gutman et al. [14] obtained a relation between a vertex-degree-based topological index and its energy. In this paper, we give various lower and upper bounds for the energy of graphs in terms of some topological indices of graphs as the first general multiplicative Zagreb index, the general Randić index, the general zeroth-order Randić index, the redefined Zagreb index, and the atom-bond connectivity index.

2 Preliminaries and Known Results

In this section, we recall some well-known results from chemical graph theory that will be used in the proofs of the upcoming sections. In [1], the authors established a relation between the Randić index and the diameter of a given graph.

Observation 1

[1] For any connected graph G on $n \ge 3$ vertices with Randić index R(G) and diameter D(G), it holds

$$\begin{aligned} R(G)D(G)\le \frac{(n - 1)(n - 3 +2\sqrt{2})}{2}. \end{aligned}$$

The next result gives a relationship between the Randić index R(G) and the maximum degree $\Delta (G)$ of a given graph. This result is published in [1].

Observation 2

[1] For any connected graph G on $n \ge 3$ vertices with Randić index R(G) and maximum degree $\Delta (G)$,

$$\begin{aligned} R(G)\Delta (G)\le \frac{n(n - 1)}{2}. \end{aligned}$$

The following result appears in [1] as well.

Observation 3

[1] For any connected graph G on $n \ge 3$ vertices with Randić index R(G) and girth g(G),

$$\begin{aligned} R(G)g(G)\le \frac{n^{2}}{2}. \end{aligned}$$

In [21], the authors give a relation between the Randić index and the algebraic connectivity of a given graph.

Observation 4

[21] For any connected graph G on $n \ge 3$ vertices with Randić index R(G) and algebraic connectivity $\alpha$,

$$\begin{aligned} R(G)\le \frac{\alpha \left( n-3+2\sqrt{2}\right) }{4\left( 1-\cos \frac{\pi }{n}\right) }. \end{aligned}$$

The first result concerns the energy of the graph in terms of its order and size. This result is given the following upper bound obtained in 1971 by McClelland [22]:

$$\begin{aligned} \mathcal {E}(G)\le \sqrt{2mn}. \end{aligned}$$

(2)

In [20], the following theorem is proved.

Theorem 1

[20] Let G be a non-regular n-vertex graph without isolated vertices and $\alpha \ne 0$. Then

$$\begin{aligned} ^{0}R_{2\alpha +1}(G)\le 2R_\alpha (G)-\left( ^{0}R_{\alpha }(G)\right) ^2+n \left( ^{0}R_{2\alpha }(G)\right) . \end{aligned}$$

In this paper, we apply the following two algebraic inequalities.

Lemma 1

[17] Let a and b $(a \ge b)$ be two non-negative real numbers. Then

$$\begin{aligned} \frac{1}{\sqrt{2}}\left( \sqrt{a}+\sqrt{b}\right) \le \sqrt{a+b}\le \sqrt{a}+\left( \sqrt{2}-1 \right) \sqrt{b}. \end{aligned}$$

Lemma 2

[26] Let a and b be positive numbers. Then

$$\begin{aligned} \left( \frac{\sqrt{a}+\sqrt{b}}{2}\right) \left( \sqrt{\frac{a+b}{2}}\right) \le \sqrt{\frac{a^{2}+b^{2}}{2}}. \end{aligned}$$

The following lemma plays a key role in this paper.

Lemma 3

[2] For a graph G and a vertex $v_i \in V(G)$,

$$\begin{aligned} \mathcal {E}(G)\le \sum _{i=1}^{n}\sqrt{d_i} \end{aligned}$$

with equality if and only if the connected component containing $v_i$ is isomorphic to $S_n$ and $v_i$ is its center.

Lemma 4

[11] If G is a non-empty graph with maximum degree $\Delta$, then $\lambda _1 \ge \sqrt{\Delta }$ with equality if and only if G is $\frac{n}{2} K_2$.

3 Main Results

In this section, we establish new relations between the energy of graphs and some well-known topological indices. The first result gives a relation between the energy and the Narumi-Katayama index of graphs.

Theorem 2

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

$$\begin{aligned} \mathcal {E}(G)\le \sqrt{2m( n-1)+n\left( NK(G)\right) ^{\frac{1}{n}} }. \end{aligned}$$

(3)

The equality holds if and only if $G\cong K_2.$

Proof

We use the following well-known inequality published in [16]. For non-negative numbers $a_1, a_2, \dots , a_n$, it holds

$$\begin{aligned} n \left( \frac{1}{n} \sum _{i=1}^{n}a_i - \left( \prod _{i=1}^{n}a_i\right) ^{\frac{1}{n}} \right) \le n\sum _{i=1}^{n}a_i-\left( \sum _{i=1}^{n}\sqrt{a_i}\right) ^{2}. \end{aligned}$$

(4)

Setting $a_i = d_i$, the inequality (4) becomes

$$\begin{aligned} n \left( \frac{1}{n} \sum _{i=1}^{n}d_i - \left( \prod _{i=1}^{n}d_i\right) ^{\frac{1}{n}} \right) \le n\sum _{i=1}^{n}d_i-\left( \sum _{i=1}^{n}\sqrt{d_i}\right) ^{2}. \end{aligned}$$

Hence

$$\begin{aligned} \sum _{i=1}^{n}\sqrt{d_i}\le \sqrt{2m( n-1)+n\left( \prod _{i=1}^{n}d_i\right) ^{\frac{1}{n}} }. \end{aligned}$$

Therefore, by Lemma 3 we get

$$\begin{aligned} \mathcal {E}(G)\le \sum _{i=1}^{n}\sqrt{d_i}\le \sqrt{2m( n-1)+n\left( \prod _{i=1}^{n}d_i\right) ^{\frac{1}{n}} }. \end{aligned}$$

If $G\cong K_2$ it is easy to check that the equality in (3) holds. Conversely, if the equality in (3) holds, according to the above argument, the equality in (4) holds. Thus $d_{1}=d_{2}=\ldots =d_{n}$, that is, G is a k-regular graph. In this case $\sum _{i=1}^{n}\sqrt{d_{i}}=n\sqrt{k}=\sqrt{2mn}.$ In order to get an equality in $\mathcal {E}(G)\leqslant \sum _{i=1}^{n}\sqrt{d_{i}}$, we consider those regular graphs where the equality between $\mathcal {E}(G)$ and the McClelland bound $\sqrt{2mn}$ holds. It is already known that such graphs are $\frac{n}{2} K_2.$ Since G is a connected graph, we get $G\cong K_2$. $\square$

Remark 1

Using the inequality between the arithmetic and geometric means for the numbers $d_{1}, d_{2}, \ldots , d_{n}$ we get

$$\begin{aligned} \sqrt{2m( n-1)+n\left( \prod _{i=1}^{n}d_i\right) ^{\frac{1}{n}} }\le \sqrt{2m( n-1)+n\left( \frac{d_1+d_2+\dots +d_n}{n}\right) }=\sqrt{2mn} \end{aligned}$$

that is, the bound in (3) is better than the bound in (2).

Theorem 1 has the following consequence for $\alpha =\frac{1}{2}$.

Corollary 1

Let G is a non-regular n-vertex graph without isolated vertices. Then

$$\begin{aligned} \mathcal {E}(G)\le \sqrt{2R_{1/2}(G)-M_1(G)+2mn}. \end{aligned}$$

(5)

Remark 2

From $R_{1/2}(G)=\sum _{(u,v)\in E(G)} \sqrt{d_{u}d_{v}}$ and from the inequality $2\sqrt{d_{u} d_{v}}\leqslant d_{u}+d_{v}$ we get $2R_{1/2}(G)\leqslant M_{1}(G)$. Thus

$$\begin{aligned} 2R_{1/2}(G)=\sum _{(u,v)\in E(G)} 2\sqrt{d_{u}d_{v}}\leqslant \sum _{(u,v)\in E(G)} (d_{u}+d_{v})=M_{1}(G). \end{aligned}$$

Hence

$$\begin{aligned} \mathcal {E}(G)\le \sqrt{2R_{1/2}(G)-M_1(G)+2mn}\leqslant \sqrt{2mn}. \end{aligned}$$

Therefore, the bound in (5) is better than the well-known bound $\sqrt{2mn}.$

Proposition 3

For any graph with $\delta >1$, we have

$$\begin{aligned} \mathcal {E}(G)\leqslant \frac{2m}{\sqrt{\delta }}. \end{aligned}$$

Proof

We have

$$\begin{aligned} \mathcal {E}(G)\le \sum _{u\in V(G)} {\sqrt{d_u}}= \sum _{u\in V(G)} d_u \frac{1}{\sqrt{d_u}}\le \frac{2m}{\sqrt{\delta }}. \end{aligned}$$

(6)

That is, for $\delta > 1$, the bound in (6) is better than the well-known upper bound 2m. $\square$

In the next proposition we reprove the bound $\mathcal {E}(G)\leqslant 2\sqrt{\Delta }$ proven in [28].

Proposition 4

Let G be a connected graph with n vertices and maximum degree $\Delta .$ Then

$$\begin{aligned} \mathcal {E}(G)\leqslant 2\sqrt{\Delta }R(G). \end{aligned}$$

Proof

We have

$$\begin{aligned} 2R(G)&=2 \sum _{uv\in E(G)} d_u^{-1/2}d_v^{-1/2}\\ {}&= \sum _{u\in E(G)}d_u^{-1/2} \sum _{v\in N(u)}d_v^{-1/2}\\&\ge \sum _{u\in E(G)}d_u^{-1/2} \sum _{v\in N(u)}\Delta ^{-1/2}= \sum _{u\in E(G)}d_u^{-1/2} d_u\Delta ^{-1/2}\\&= \sum _{u\in E(G)}d_u^{1/2}\Delta ^{-1/2}\geqslant \frac{\mathcal {E}(G)}{\sqrt{\Delta }}. \end{aligned}$$

$\square$

The next theorem reveals a connection between the energy and the redefined first general Zagreb index of graphs.

Theorem 5

Let G be a connected graph of order $n > 2$. Then

$$\begin{aligned} \mathcal {E}(G)\le \sqrt{2} ReZg_{1/2}. \end{aligned}$$

(7)

The equality is attained if and only if $G\cong K_2.$

Proof

By using Lemmas 1 and 3, we have

$$\begin{aligned} \mathcal {E}(G)\le & {} \sum _{u\in V(G)}\sqrt{d_u}=\sum _{uv\in {E}(G) }\left( \frac{1}{\sqrt{d_u}}+ \frac{1}{\sqrt{d_v}}\right) =\sum _{uv\in {E}(G) } \frac{\sqrt{d_u}+\sqrt{d_v}}{\sqrt{d_ud_v}}\nonumber \\\le & {} \sum _{uv\in {E}(G) } \frac{\sqrt{2}\sqrt{d_u+d_v}}{\sqrt{d_ud_v}} =\sum _{uv\in {E}(G) } \sqrt{\frac{2\left( d_u+d_v \right) }{d_ud_v}}. \end{aligned}$$

(8)

This implies the result stated in the theorem. If $G\cong K_2$ it is easy to check that the equality in (7) holds. Conversely, if the equality in (7) holds, then the equality in (8) holds, which is possible only if $d_{u}=d_{v}.$ Thus G is a regular graph. As before, we proved that the unique regular graph which satisfies the identity $\mathcal {E}(G)=\sum _{v\in V(G)}\sqrt{d_{v}}$ is $K_{2}.$ $\square$

Remark 3

By Lemma 1 we have

$$\begin{aligned} \sqrt{2} ReZg_{1/2}&=\sum _{uv\in {E}(G) } \sqrt{\frac{2\left( d_u+d_v \right) }{d_ud_v}}\\ {}&\le \sum _{uv\in {E}(G) } \frac{\sqrt{2}\left( \sqrt{d_u}+ (\sqrt{2} - 1)\sqrt{d_v}\right) }{\sqrt{d_ud_v}}\\&\le \sum _{uv\in {E}(G) } \frac{\sqrt{2}\left( \sqrt{\Delta }+ (\sqrt{2} - 1)\sqrt{\Delta }\right) }{\sqrt{d_ud_v}}\\ {}&=2\sqrt{\Delta }R(G). \end{aligned}$$

Thus, the bound in (7) is better than the bound in Proposition 5.

The next result concerns the energy of graphs in terms of the Randić index and the maximum and the minimum degree of G.

Theorem 6

Let G be a connected graph with maximum degree $\Delta$ and minimum degree $\delta$. If $|\sqrt{d_v}-\sqrt{d_u}|\geqslant 1$ for each edge $(u,v)\in E(G),$ then

$$\begin{aligned} \mathcal {E}(G)\le \left( \Delta -\delta \right) R(G). \end{aligned}$$

(9)

Proof

From $\left( \frac{1}{\sqrt{d_u}} +\frac{1}{\sqrt{d_v}}\right) \left( \frac{1}{\sqrt{d_u}} -\frac{1}{\sqrt{d_v}}\right) =\frac{1}{{d_u}} -\frac{1}{{d_v}}$ we get

$$\begin{aligned} \frac{1}{\sqrt{d_u}} +\frac{1}{\sqrt{d_v}}=\frac{d_v-d_u}{\left( \sqrt{d_v}-\sqrt{d_u}\right) \sqrt{d_ud_v} }. \end{aligned}$$

Since for any edge $uv\in E(G)$ it holds $\frac{1}{\sqrt{d_v}-\sqrt{d_u}} \le 1$ we get

$$\begin{aligned} \mathcal {E}(G)&\leqslant \sum _{u\in V(G)}\sqrt{d_u}\\&=\sum _{uv\in E(G)} \left( \frac{1}{\sqrt{d_u}} +\frac{1}{\sqrt{d_v}}\right) \\&=\sum _{uv\in E(G)}\frac{d_v-d_u}{\left( \sqrt{d_v}-\sqrt{d_u}\right) \sqrt{d_ud_v} }\\&\le \sum _{uv\in E(G)}\frac{|d_v-d_u|}{ \sqrt{d_ud_v} }\\&\le \left( \Delta -\delta \right) R(G). \end{aligned}$$

This implies the result stated in the theorem. $\square$

The next theorem reveals a connection between the energy and the atom-bond connectivity (ABC) index of graphs.

Theorem 7

Let G be a graph of order n with no isolated vertices. If $\delta \ge 2$, then

$$\begin{aligned} \mathcal {E}(G)\le 2ABC(G). \end{aligned}$$

Proof

By using the definitions and Lemma 3, we have

$$\begin{aligned} \mathcal {E}(G)&\le \sum _{v\in V(G) }\sqrt{d_i}\\ {}&=\sum _{uv\in {E}(G) }\left( \frac{1}{\sqrt{d_u}}+ \frac{1}{\sqrt{d_v}}\right) \quad (as\,\, d_u,d_v \ge \delta \ge 2)\\&\le \sum _{uv\in {E}(G) }\left( \sqrt{\frac{1}{d_u}+\frac{d_u-2}{d_ud_v}}+\sqrt{\frac{1}{d_v}+\frac{d_v-2}{d_ud_v}}\right) \\&=2\sum _{uv\in {E}(G) }\sqrt{\frac{1}{d_u}+\frac{1}{d_u}-\frac{2}{d_ud_v}}\\&=2ABC(G). \end{aligned}$$

$\square$

In the next theorem, we determine an upper bound for the energy of graphs in terms of the maximum eigenvalues $\lambda _1$ and the inverse degree of G.

Theorem 8

Let G be a non-trivial connected graph of order n and maximum eigenvalue $\lambda _1$. Then

$$\begin{aligned} \mathcal {E}(G)\le \lambda _1\sqrt{nID(G)}. \end{aligned}$$

(10)

The equality is attained if and only if $G\cong K_2.$

Proof

For $1 \le i \le n$ let $a_i$ and $b_i$ be real numbers. In this proof we use Cauchy-Schwarz inequality (see [16]):

$$\begin{aligned} \left( \sum _{i= 1}^{n}a_ib_i\right) ^2 \le \left( \sum _{i= 1}^{n}a_i^{2} \right) \left( \sum _{i= 1}^{n}b_i^{2} \right) . \end{aligned}$$

(11)

If we take $a_i=d_i$ and $b_i=\frac{1}{\sqrt{d_i}}$ in inequality (11), we have

$$\begin{aligned} \left( \sum _{i= 1}^{n}\sqrt{d_i}\right) ^2 \le \left( \sum _{i= 1}^{n}d_i^{2} \right) \left( \sum _{i= 1}^{n}\frac{1}{d_i} \right) . \end{aligned}$$

(12)

In [11], it is proved that

$$\begin{aligned} \lambda _1\ge \sqrt{\frac{\sum _{i= 1}^{n}d_i^{2}}{n}}. \end{aligned}$$

(13)

From the above inequality and (12) we obtain (10).

If $G\cong K_2$ it is easy to check that the equality in (10) holds. Conversely, if the equality in (10) holds, then the equality in Cauchy-Schwarz inequality holds, that is, $\frac{d_{i}}{\frac{1}{\sqrt{d_{i}}}}=\frac{d_{j}}{\frac{1}{\sqrt{d_{j}}}}$ for each $i\ne j.$ Hence $d_{i}=d_{j}$, i.e., G is a regular graph. In this case holds equality in (13), since $\lambda _{1}=k.$ We already prove that the unique regular graph which satisfies $\mathcal {E}(G)=\sum _{i=1} ^{n}\sqrt{d_{i}}$ is the complete graph on two vertices $K_{2}.$ $\square$

Now we present a relationship between the energy and the general zeroth-order Randić index of graphs.

Theorem 9

Let G be a graph of order n, maximum degree $\Delta$ and minimum degree $\delta$. Then

$$\begin{aligned} \mathcal {E}(G)\le {^0}R_{1/4}(G)[\delta ^{1/4}+\Delta ^{1/4}]-n\delta ^{1/4}\Delta ^{1/4}. \end{aligned}$$

The equality is attained if and only if $G\cong K_2.$

Proof

Let $a_1,a_2,\dots , a_n$ be real numbers such that $a \le a_i \le A$ for all $1 \le i \le n$. Let $\mu =\frac{\sum _{i= 1}^{n}a_i}{n}$. Then from [6], the following inequality occurs:

$$\begin{aligned} \sum _{i= 1}^{n}a_i^{2}\le n\left( {\mu }[A+a]-Aa\right) . \end{aligned}$$

(14)

Let $a_i =d_i^{1/4}$, $\mu =\frac{\sum _{i= 1}^{n}d_i^{1/4}}{n}$ and $\delta ^{1/4}\le a_i\le \Delta ^{1/4}$. The inequality (14) is equivalent to

$$\begin{aligned} \sum _{i= 1}^{n}\sqrt{d_i}\le & {} n\left( \frac{\sum _{i= 1}^{n}d_i^{1/4}}{n}[\delta ^{1/4}+\Delta ^{1/4}]-\delta ^{1/4}\Delta ^{1/4}\right) \\ {}= & {} {^0}R_{1/4}(G )[\delta ^{1/4}+\Delta ^{1/4}]-n\delta ^{1/4}\Delta ^{1/4}. \end{aligned}$$

Now Lemma 3 implies the required result. By the same argument as before, we can prove that the equality holds if and only if $G\cong K_2.$ $\square$

In the next result, we give a relation between the energy, the inverse degree and the general zeroth-order Randić index of graphs.

Theorem 10

If G is a graph without isolated vertices, of order n, size m, minimum degree $\delta$ and maximum degree $\Delta$, then

$$\begin{aligned} \mathcal {E}(G)\leqslant \sqrt{(2m-n)(n-ID(G))}+^{0}R_{-1/2}(G). \end{aligned}$$

(15)

The equality is attained if and only if $G\cong K_2.$

Proof

Let $w_1,w_2, \ldots ,w_n$ be non-negative real numbers (weights). We use the weighted version of the Cauchy-Schwarz inequality

$$\begin{aligned} \sum _{i= 1}^{n}w_ia_i^{2}\sum _{i= 1}^{n}w_ib_i^{2}\ge \left( \sum _{i= 1}^{n}w_ia_ib_i\right) ^{2}. \end{aligned}$$

Let $w_{i}=d_{i}-1, a_{i}=1$ and $b_{i}=\frac{1}{\sqrt{d_{i}}},$ for each $i=1,2,\ldots , n.$ Thus the above inequality is equivalent to

$$\begin{aligned} \sum _{i=1}^{n}(d_{i}-1)\sum _{i=1}^{n}(d_{i}-1)\cdot \frac{1}{d_{i}} \geqslant \left( \sum _{i=1}^{n}\frac{d_{i}-1}{\sqrt{d_{i}}}\right) ^{2} \end{aligned}$$

and this if and only if

$$\begin{aligned} (2m-n)(n-ID(G))\geqslant \left( \sum _{i=1}^{n} \sqrt{d_{i}}-{^0}R_{-1/2}(G)\right) ^{2}. \end{aligned}$$

From $\mathcal {E}(G)\leqslant \sum _{i=1}^{n}\sqrt{d_{i}}$ we get

$$\begin{aligned} \mathcal {E}(G)\leqslant \sqrt{(2m-n)(n-ID(G))}+^{0}R_{-1/2}(G). \end{aligned}$$

If $G\cong K_2$ it is easy to check that the equality in (15) holds. On the other hand, since the weighted Cauchy-Schwarz inequality becomes equality when $\frac{a_{i}}{b_{i}}=\sqrt{d_{i}}$ is a constant for each i, we get that $d_{1}=d_{2}=\cdots =d_{n}$, that is, G is a regular graph. As before we conclude that $G\cong K_2$. $\square$

The next result gives a relationship between the energy, the general zeroth-order Randić index and the first general multiplicative Zagreb index of graphs.

Theorem 11

Let G be a graph of order n. Then

$$\begin{aligned} \mathcal {E}(G)\le \left( {^0}R_{1/4}(G)\right) ^{2}-n(n-1)\left( P_1^{1/2} (G)\right) ^{1/n}. \end{aligned}$$

The equality is attained if and only if $G\cong K_2.$

Proof

Let $a_{1},a_{2}, \ldots , a_{n}$ be positive real numbers. We apply the next inequality proved in [19]

$$\begin{aligned} \left( \sum _{i= 1}^{n} \sqrt{a_i} \right) ^{2}\ge \sum _{i= 1}^{n} a_i+n(n-1)\left( \prod _{i= 1}^{n} {a_i} \right) ^{1/n}. \end{aligned}$$

(16)

Setting $a_i = \sqrt{d_i}$ for $i=1,\ldots ,n$, the inequality (16) becomes

$$\begin{aligned} \left( \sum _{i= 1}^{n} d_i^{1/4} \right) ^{2}\ge \sum _{i= 1}^{n} \sqrt{d_i}+n(n-1)\left( \prod _{i= 1}^{n} {\sqrt{d_i}} \right) ^{1/n} \end{aligned}$$

that is

$$\begin{aligned} \sum _{i= 1}^{n} \sqrt{d_i}\le \left( {^0}R_{1/4}(G)\right) ^{2}-n(n-1)\left( P_1^{1/2} (G)\right) ^{1/n}. \end{aligned}$$

Lemma 3 implies the result stated in the theorem. By the same argument as before, we can prove that equality holds if and only if $G\cong K_2.$ $\square$

Let G be a graph without isolated vertices. In the next theorem, we determine an upper bound on the energy of a graph in terms of its size, minimum degree, maximum degree, and general Randić index.

Theorem 12

Let G be a graph of size m, with no isolated vertices, maximum degree $\Delta$ and minimum degree $\delta$. Then

$$\begin{aligned} \mathcal {E}(G)\le m \left( \frac{1}{\root 4 \of {\delta }}- \frac{1}{\root 4 \of {\Delta }}\right) ^{2}+2R_{-1/4}(G). \end{aligned}$$

(17)

The equality is attained if and only if $G\cong K_2.$

Proof

For $e=(u,v)\in E(G)$ we use the following identity

$$\begin{aligned} \left( \frac{1}{\sqrt{d_u}}+ \frac{1}{\sqrt{d_v}}\right) = \left( \frac{1}{\root 4 \of {d_u}}- \frac{1}{\root 4 \of {d_v}}\right) ^{2}+\frac{2}{\root 4 \of {d_ud_v}}. \end{aligned}$$

(18)

Hence, by the definitions, we get

$$\begin{aligned} \mathcal {E}(G)\leqslant & {} \sum _{uv\in {E}(G) }\left( \frac{1}{\sqrt{d_u}}+ \frac{1}{\sqrt{d_v}}\right) \nonumber \\= & {} \sum _{uv\in {E}(G) }\left( \frac{1}{\root 4 \of {d_u}}- \frac{1}{\root 4 \of {d_v}}\right) ^{2}+\sum _{uv\in {E}(G) }\frac{2}{\root 4 \of {d_ud_v}} \end{aligned}$$

(19)

$$\begin{aligned}\le & {} \sum _{uv\in {E}(G) }\left( \frac{1}{\root 4 \of {\delta }}- \frac{1}{\root 4 \of {\Delta }}\right) ^{2}+\sum _{uv\in {E}(G) }\frac{2}{\root 4 \of {d_ud_v}}\\\nonumber= & {} m \left( \frac{1}{\root 4 \of {\delta }}- \frac{1}{\root 4 \of {\Delta }}\right) ^{2}+2R_{-1/4}(G). \end{aligned}$$

(20)

$\square$

Note that $\left( \frac{1}{\sqrt{d_u}}+ \frac{1}{\sqrt{d_v}}\right) = \left( \frac{1}{\root 4 \of {d_u}}+ \frac{1}{\root 4 \of {d_v}}\right) ^{2}-\frac{2}{\root 4 \of {d_ud_v}}.$ By this identity we get the following result.

Corollary 2

Let G be a graph of size m with no isolated vertices and minimum degree $\delta$. Then

$$\begin{aligned} \mathcal {E}(G)\le \frac{4m}{\sqrt{\delta }}-2R_{-1/4}(G). \end{aligned}$$

In the next two theorems, we provide a relationship between the energy and the general zeroth-order Randić index of graphs.

Theorem 13

If G is a non-trivial graph of size m, then

$$\begin{aligned} \mathcal {E}(G)\le \frac{2m {^0}R_{3/4}(G)+{^0}R_{1/4}(G)F(G)}{2M_1(G)}. \end{aligned}$$

The equality is attained if and only if $G\cong K_2.$

Proof

If $a_i$ and $b_i$ are nonnegative real numbers, then the following inequality holds ([8], p. 4).

$$\begin{aligned} \frac{1}{2}\left[ \sum _{i= 1}^{n}a_i^{3} \sum _{i= 1}^{n}b_i+\sum _{i= 1}^{n}a_i \sum _{i= 1}^{n}b_i^{3}\right] \ge \sum _{i= 1}^{n}a_i^{2}\sum _{i= 1}^{n}b_i^{2}. \end{aligned}$$

(21)

For $a_i=d_i^{1/4}$ and $b_i=d_i$, the inequality (21) becomes

$$\begin{aligned} \frac{1}{2}\left[ \sum _{i= 1}^{n}d_i^{3/4} \sum _{i= 1}^{n}d_i+\sum _{i= 1}^{n}d_i^{1/4} \sum _{i= 1}^{n}d_i^{3}\right] \ge \sum _{i= 1}^{n}\sqrt{d_i}\sum _{i= 1}^{n}d_i^{2} \end{aligned}$$

from where we get

$$\begin{aligned} \sum _{i= 1}^{n}\sqrt{d_i}\le \frac{2m {^0}R_{3/4}(G)+{^0}R_{1/4}(G)F(G)}{2M_1(G)}. \end{aligned}$$

Lemma 3 leads to the desired bound. By the same argument as before, we can prove that equality holds if and only if $G\cong K_2.$ $\square$

Theorem 14

Let G be a graph without isolated vertices, of order n, minimum degree $\delta$ and maximum degree $\Delta$. Then

$$\begin{aligned} \mathcal {E}(G)\le ( \root 4 \of {\delta } + \root 4 \of {\Delta }) {^0}R_{1/4}(G)-n\root 4 \of {\delta \Delta }. \end{aligned}$$

(22)

The equality is attained if and only if $G\cong K_2.$

Proof

Let $x_1, x_2,\dots , x_n$ and $y_1, y_2, \ldots , y_n$ be real numbers such that there exist real constants a and A so that for each $i = 1, 2, \ldots , n$ holds $ax_i \le y_i \le Ax_i$. Then the following inequality is valid (see [7])

$$\begin{aligned} \sum _{i=1}^{n}y_i^{2}+Aa\sum _{i=1}^{n}x_i^{2}\le (a + A)\sum _{i=1}^{n}x_iy_i. \end{aligned}$$

(23)

For $y_i= d^{1/4}_i, x_i= 1, a= \root 4 \of {\delta }$ and $A = \root 4 \of {\Delta }, i = 1, 2, \ldots , n$ the inequality (23) becomes

$$\begin{aligned} \sum _{i=1}^{n}\sqrt{d_i}+ \root 4 \of {\delta \Delta }\sum _{i=1}^{n}1 \le ( \root 4 \of {\delta } + \root 4 \of {\Delta })\sum _{i=1}^{n}d_{i}^{1/4}, \end{aligned}$$

that is,

$$\begin{aligned} \sum _{i=1}^{n}\sqrt{d_i}\le ( \root 4 \of {\delta } + \root 4 \of {\Delta }) {^0}R_{1/4}(G)-n\root 4 \of {\delta \Delta }. \end{aligned}$$

The last inequality leads to the desired bound. By the same argument as before, we can prove that the equality holds if and only if $G\cong K_2.$

$\square$

The following result gives a relation between the energy, the first Zagreb index and the maximum degree of G.

Theorem 15

Let G be a non-trivial connected graph of order n and maximum degree $\Delta$. Then

$$\begin{aligned} \mathcal {E}(G)\le M_1(G)+ \sqrt{\Delta }-\Delta ^{2}-\ln \left( \frac{NK(G)}{\Delta }\right) . \end{aligned}$$

The equality is attained if and only if $G\cong K_2.$

Proof

We consider the following function for $x>0$

$$\begin{aligned} g(x) =x^{2}-\sqrt{x}-\ln (x). \end{aligned}$$

The first derivative of g(x) is $g'(x) =2x-\frac{1}{2\sqrt{x}}-\frac{1}{x}.$ It is easy to note that for $x \ge 1$, g(x) is increasing, thus $g(x)=x^{2}-\sqrt{x}-\ln (x)\geqslant 1^{2}-\sqrt{1}-\ln (1)=0.$ Hence $\sqrt{x}\le x^{2}-\ln (x)$. Using this result and Lemma 3, we get

$$\begin{aligned} \mathcal {E}(G)&\le \sum _{i=1}^{n}\sqrt{d_i}\\ {}&=\sqrt{d_1}+ \sum _{i=2}^{n}\sqrt{d_i}\le \sqrt{d_1}+ \sum _{i=2}^{n}\left( d_i^{2}-\ln (d_i)\right) \\&= M_1(G)+ \sqrt{\Delta }-\Delta ^{2}-\ln \left( \prod _{i=2}^{n}d_i \right) \\ {}&=M_1(G)+ \sqrt{\Delta }-\Delta ^{2}-\ln \left( \frac{NK(G)}{\Delta } \right) . \end{aligned}$$

This completes the proof of the theorem. By the same argument as before, we can prove that equality holds if and only if $G\cong K_2.$ $\square$

4 Applications

In this section, we present several results which relate the energy of the graph and its diameter, girth, algebraic connectivity and its radius. The results are based on Observation 1–4.

By $\mathcal {E}(G)\leqslant 2\sqrt{\Delta }R(G)$ and Observation 1, we obtain a relation between the energy and the diameter of the graph.

Lemma 5

Let G be a connected graph of order $n \ge 3$, maximum degree $\Delta$ and diameter D. Then

$$\begin{aligned} \mathcal {E}(G)\le \frac{\sqrt{\Delta }(n - 1)(n - 3 +2\sqrt{2})}{D(G)}. \end{aligned}$$

Based on Observation 2, we obtain a result that gives a relation between the energy, the maximum and the minimum degree of graphs.

Lemma 6

Let G be a connected graph of order $n \ge 3$, maximum degree $\Delta$ and minimum degree $\delta$. Then

$$\begin{aligned} \mathcal {E}(G)\le \frac{n(n - 1)}{2}-\frac{\delta n(n - 1)}{2\Delta }. \end{aligned}$$

By Observation 3 we obtain a relation between the energy and the girth of a given graph.

Lemma 7

Let G be a connected graph of order $n \ge 3$, maximum degree $\Delta$ and girth g. Then

$$\begin{aligned} \mathcal {E}(G)\le \frac{\sqrt{\Delta }n^{2}}{g(G)}. \end{aligned}$$

By Observation 4, we obtain a relation between the energy and the algebraic connectivity of a graph.

Lemma 8

Let G be a connected graph of order $n \ge 3$, maximum degree $\Delta$ and algebraic connectivity $\alpha$. Then

$$\begin{aligned} \mathcal {E}(G)\le \frac{\sqrt{\Delta }\alpha \left( n-3+2\sqrt{2}\right) }{2\left( 1-\cos \frac{\pi }{n}\right) }. \end{aligned}$$

Lemma 9

Let G be a non-trivial connected graph of order n. Then

$$\begin{aligned} \mathcal {E}(G)\le n\sqrt{n-r}. \end{aligned}$$

Proof

For each vertex $v \in V (G)$, we have $d_v \le n - ecc_G(v)$. According to the definition of the energy, we obtain

$$\begin{aligned} \mathcal {E}(G)\le \sum _{v \in V (G)}\sqrt{d_v}\le \sum _{v \in V (G)}\sqrt{n - ecc_G(v)}\le n\sqrt{n-r}. \end{aligned}$$

$\square$