# Energy of Nonsingular Graphs: Improving Lower Bounds

Authors: Hajar Shooshtari; Jonnathan Rodriguez; Akbar Jahanbani; Abbas Shokri
Journal: Journal of Mathematics
 (2021)

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## Abstract

LetG be a simple graph of order n and A be its adjacency matrix. Let λ1≥λ2≥…≥λn be eigenvalues of matrix A. Then, the energy of a graph G is defined as εG=∑i=1nλi. In this paper, we will discuss the new lower bounds for the energy of nonsingular graphs in terms of degree sequence, 2-sequence, the first Zagreb index, and chromatic number. Moreover, we improve some previous well-known bounds for connected nonsingular graphs.


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## Body

## 1. Introduction

In this paper, we assume thatG is a simple graph and that VG and EG are the vertex set and the edge set so that VG=n and EG=m. Let di be the degree of vertex vi. For convenience, we assume here that Kn and Ka,b are the complete graph and the complete bipartite graph, respectively.Graph coloring is a way of coloring the vertices of a graph such that no two adjacent vertices are of the same color; this is called vertex coloring. The smallest number of colors needed to color a graphG is called its chromatic number of G, denoted by χG.The sum of the degrees of the vertices adjacent tovi is call the 2-degree of vi and denoted by hi. The average degree is 2 degree, and we denote by hi/di the average degree of vi. The first Zagreb of G, introduced in [1], is defined as follows:(1)M1G=∑vi∈Vdvi2.Assuming thatλ1≥λ2≥…≥λn is eigenvalues of adjacency matrix A, we know that(2)detA=∏i=1nλi.IfdetA=0, we call G singular, otherwise we call it nonsingular.According to the eigenvalues of the adjacency matrix, the energy of a graph is defined as follows:(3)εG=∑i=1nλi.Graph energy was first used in chemistry to approximate the energy ofπ-electron of a molecule [2, 3].Liu et al. [4] derived some new bounds for the energy. Filipovski and Jajcay [5], derived some of the bounds for the energy. Das and Gutman [6] discussed bounds for the energy and improved some of the bounds. In 2017, Jahanbani [7] obtained some of the lower bounds for the energy. In 2018, Jahanbani [8] obtained some of the upper bounds for the energy and improved well-known bounds. In 2020, Filipovski and Jajcay [5] derived some of lower bounds for the energy. In 2021, Filipovski and Jajcay [5] obtained new bounds for the energy. In this paper, we continue this discussion by obtaining new bounds for the energy of nonsingular connected graph and improving some important bounds.The oldest bounds are discovered by McClelland [9–12]. Bounds have been favored by researchers in the mathematical sciences, see [5, 6, 8, 13–17].McClelland, in [12], obtained the next result:(4)εG≥2m+nn−1detA2/n1/2.The proof of the following bound can be found in [18]:(5)εG≥2m.The next result is obtained by Das et al. in [19]:(6)εG≥2mn+n−1+lndetA−ln2mn.


## 2. Preliminaries

In this section, we recall some of the results that we will need to prove the main results.It is straightforward to demonstrate the following two results.Lemma 1.
Consider functionf1 as follows:(7)f1y=y−1−lny.Then, functionsf1y are increasing for y≥1 and decreasing for 0<y≤1.Lemma 2.
Functiongy=y+n−1+lndetA−lny is an increasing function on 1,n.Lemma 3 (see [20]).
For a connected graphG with n vertices and m edges, we have(8)λ1≥2mn.Lemma 4 (see [21]).
For a nonempty graph, we have(9)λ1≥∑i=1nhi2∑i=1ndi2.Lemma 5 (see [22, 23]).
For a connected graph withn vertices, we have(10)λ1≥M1Gn.Lemma 6 (see [20]).
For a connected graph with chromatic numberχ, we have(11)λ1≥χ−1.Lemma 7 (see [24]).
SupposeG be a graph with n≥2 vertices; then,(12)λ1≥12maxj<idi+dj+di−dj2+4cij2,where cij is the number of common neighbours of i and j.Lemma 8 (see [24]).
SupposeG be a graph with n vertices; then,(13)λ1≥1n∑i=1ndi+2∑i>jcij24,where cij is the number of common neighbours of i and j.Lemma 9 (see [25]).
LetG be a graph; then, it has only one distinct eigenvalue if and only if G is an empty graph and G has 2 distinct eigenvalues λ1>λ2 with multiplicities s1 and s2 if and only if G is the direct sum of m1 complete graphs of order λ1+1. Also, λ2=−1 and s2=s1λ1.


## 3. Lower Bounds for the Energy of Nonsingular Graphs

In this section, we present new lower bounds for energy of a nonsingular graphG.Theorem 1.
LetG be a nonempty and nonsingular graph with n vertices and m edges. Then,(14)εG≥M1Gn+n−1+lndetA−lnM1Gn.Equality holds if and only ifG≅Kn.Proof.
Note thatG is nonsingular; hence, we have λi>0, for i=1,2,…,n. From Lemma 1, we have f1y≥f11=0; therefore,(15)y≥1+lny,where y>0, and the equality holds if and only if y=1. By applying the definition of energy, we can write(16)εG=λ1+∑i=2nλi≥λ1+n−1+∑i=2nlnλi=λ1+n−1+ln∏i=2nλi=λ1+n−1+lndetA−lnλ1.
By Lemma5, we have λ1≥M1G/n. From Lemma 2, we obtain(17)gλ1≥gM1Gn=M1Gn+n−1+lndetA−lnM1Gn.
From the above result and equality (16), we get our result.
Now, to prove the second part of the theorem, ifG≅Kn, it can be easily seen that the equality in Theorem 1 holds. Conversely, if the equality in Theorem 1 holds, then, by Lemma 5, we obtain(18)λ1=M1Gn.
Note thatG is a nonempty graph; by using Lemma 9, we know that the graph G has at least two distinct eigenvalues. Hence, we continue the proof with the following two cases.Case 1.
Letλ1=λ2=…=λn.
Note that∑i=1nλi2=2m holds directly for any graph with m edges. Hence, we have ∑i=2nλi2=2m−λ12. So, we have λ1=λi=2m−λ12/n−1, since G has at least two distinct eigenvalues for 2≤i≤n. By inequality (15), we have y≥1+lny, and since the equality holds for y=1, we directly have that the absolute values of λ2,…λn is 1. Hence, 2m=n and also λ1=λ2=…=λn=1. By applying Lemma 9, we obtain that s2=s1λ1,λ1=1; then, s1=s2. Thereby, we obtain that λ1=1 has multiplicity n/2, and λi=−1 has multiplicity n/2 for 2≤i≤n. Therefore, G is the direct sum of s1=n/2 complete graphs of order λ1+1=2. Therefore, G≅n/2K2, which we see is in contradiction with the nonsingular graph.Case 2.
The absolute value of all eigenvalues ofG is not equal. Then, G has 2 distinct eigenvalues with different absolute values. Similar to Case 1, we have that the absolute values of λ2,…λn is 1. Since, ∑i=1nλi=0 and λi=−1 for 2≤i≤n, then we have λ1=n−1. Hence, λ1 has multiplicity 1 and λi=−1 has multiplicity n−1. By Lemma 9, G is the direct sum of a complete graph of order λ1+1=n. In other words, G≅Kn.Using the technique to demonstrate Theorem1, we get the next result.Theorem 2.
For any nonempty and nonsingular connected graphG with n vertices and chromatic number χ, we have(19)εG≥χ−1+n−1+lndetA−lnχ−1.Equality in (19) holds if and only if G≅Kn.Theorem 3.
For any nonempty and nonsingular graphG with n vertices, we have(20)εG≥∑i=1nhi2∑i=1ndi2+n−1+lndetA−ln∑i=1nhi2∑i=1ndi2.Proof.
Note thatG is nonsingular; hence, we have λi>0, for i=1,2,…,n. Thus,(21)detA=∏i=1nλi>0.
By equality (16), we can write(22)εG=λ1+∑i=2nλi≥λ1+n−1+∑i=2nlnλi=λ1+n−1+ln∏i=2nλi=λ1+n−1+lndetA−lnλ1.
From Lemma4, we have λ1≥∑i=1nhi2/∑i=1ndi2. By Lemma 2, we can write(23)gλ1≥g∑i=1nhi2∑i=1ndi2=∑i=1nhi2∑i=1ndi2+n−1+lndetA−ln∑i=1nhi2∑i=1ndi2.
By inequality (23) and equality (22), we get our result.Theorem 4.
For any nonempty and nonsingular graph withn vertices, we have(24)EG≥12maxj<idi+dj+di−dj2+4cij2+n−1+lndetA−ln12maxj<idi+dj+di−dj2+4cij2.Proof.
With the same argument as before, we can write(25)εG=λ1+∑i=2nλi≥λ1+n−1+lndetA−lnλ1.
From Lemma7, we obtain(26)λ1≥12maxj<idi+dj+di−dj2+4cij2.
According to the properties of functiong, we have that(27)gλ1≥g12maxj<idi+dj+di−dj2+4cij2=12maxj<idi+dj+di−dj2+4cij2+n−1+lndetA−ln12maxj<idi+dj+di−dj2+4cij2.
From the above inequality and equality (25), we obtain our result.
Similarly to Theorem4 and by using Lemma 8, we can reach the following result.Theorem 5.
LetG be a nonempty and nonsingular graph with n vertices. Then,(28)εG≥1n∑i=1ndi+2∑i>jcij24+n−1+lndetA−ln1n∑i=1ndi+2∑i>jcij24.


## 4. Improving Some of Bounds for the Energy of Connected Nonsingular Graphs

In this section, we show that the lower bounds in (14) and (20) are better than the classical bound [19] given by(29)εG≥2mn+n−1+lndetA−ln2mn,for nonsingular connected graphs. Moreover, we show that inequality (20) is better than inequality (14).Theorem 6.
The bound in (14) improves the well-known bound in (6) for all connected nonsingular graphs.Proof.
Sincegx is increasing on 1,n and since M1G/n≥2m/n≥1 (note that M1G≥4m2/n), hence, we have gM1G/n≥g2m/n, that is, the bound in (14) is better than the bound in (6).Corollary 1.
The bound in (14) improves the well-known bound in (4) for all connected nonsingular graphs.Theorem 7.
The bound in (20) improves the well-known bound in (6) for all connected nonsingular graphs.Proof.
Since the bound in (19) is always better the bound in (6), the proof relies on the same facts as in Theorem 6. Use the relation ∑i=1nhi2/∑i=1ndi2≥2m/n and g∑i=1nhi2/∑i=1ndi2≥g2m/n, that is, the bound in (20) is better than the bound in (6).Theorem 8.
The bound in (20) improves the bound in (14) for all connected nonsingular graphs.Proof.
Since(30)∑i=1nhi2∑i=1ndi2≥∑i=1nhi2n∑i=1ndi2=1n∑i=1ndi2=M1Gn,by using the properties of the g function, we can write g∑i=1nhi2/∑i=1ndi2≥gM1G/n, that is, the bound in (20) is better than the bound in (14).



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