Abstract
Let be a simple graph of order and be its adjacency matrix. Let be eigenvalues of matrix . Then, the energy of a graph is defined as . 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.
1. Introduction
In this paper, we assume that is a simple graph and that and are the vertex set and the edge set so that and . Let be the degree of vertex . For convenience, we assume here that and 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 graph is called its chromatic number of , denoted by .
The sum of the degrees of the vertices adjacent to is call the 2-degree of and denoted by . The average degree is 2 degree, and we denote by the average degree of . The first Zagreb of G, introduced in [1], is defined as follows:
Assuming that is eigenvalues of adjacency matrix , we know that
If , 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:
Graph energy was first used in chemistry to approximate the energy of -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:
The proof of the following bound can be found in [18]:
The next result is obtained by Das et al. in [19]:
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 function as follows:
Then, functions are increasing for and decreasing for .
Lemma 2. Function is an increasing function on .
Lemma 3 (see [20]). For a connected graph with vertices and edges, we have
Lemma 4 (see [21]). For a nonempty graph, we have
Lemma 5 (see [22, 23]). For a connected graph with vertices, we have
Lemma 6 (see [20]). For a connected graph with chromatic number , we have
Lemma 7 (see [24]). Suppose be a graph with vertices; then,where is the number of common neighbours of and .
Lemma 8 (see [24]). Suppose be a graph with vertices; then,where is the number of common neighbours of and .
Lemma 9 (see [25]). Let be a graph; then, it has only one distinct eigenvalue if and only if is an empty graph and has 2 distinct eigenvalues with multiplicities and if and only if is the direct sum of complete graphs of order . Also, and .
3. Lower Bounds for the Energy of Nonsingular Graphs
In this section, we present new lower bounds for energy of a nonsingular graph .
Theorem 1. Let be a nonempty and nonsingular graph with vertices and edges. Then,
Equality holds if and only if .
Proof. Note that is nonsingular; hence, we have , for . From Lemma 1, we have ; therefore,where , and the equality holds if and only if . By applying the definition of energy, we can writeBy Lemma 5, we have . From Lemma 2, we obtainFrom the above result and equality (16), we get our result.
Now, to prove the second part of the theorem, if , 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 obtainNote that is a nonempty graph; by using Lemma 9, we know that the graph has at least two distinct eigenvalues. Hence, we continue the proof with the following two cases.
Case 1. Let .
Note that holds directly for any graph with m edges. Hence, we have . So, we have , since has at least two distinct eigenvalues for . By inequality (15), we have , and since the equality holds for , we directly have that the absolute values of is 1. Hence, and also . By applying Lemma 9, we obtain that ; then, . Thereby, we obtain that has multiplicity , and has multiplicity for . Therefore, is the direct sum of complete graphs of order . Therefore, , which we see is in contradiction with the nonsingular graph.
Case 2. The absolute value of all eigenvalues of is not equal. Then, has 2 distinct eigenvalues with different absolute values. Similar to Case 1, we have that the absolute values of is 1. Since, and for , then we have . Hence, has multiplicity 1 and has multiplicity . By Lemma 9, is the direct sum of a complete graph of order . In other words, .
Using the technique to demonstrate Theorem 1, we get the next result.
Theorem 2. For any nonempty and nonsingular connected graph with vertices and chromatic number , we have
Equality in (19) holds if and only if .
Theorem 3. For any nonempty and nonsingular graph with vertices, we have
Proof. Note that is nonsingular; hence, we have , for . Thus,By equality (16), we can writeFrom Lemma 4, we have . By Lemma 2, we can writeBy inequality (23) and equality (22), we get our result.
Theorem 4. For any nonempty and nonsingular graph with vertices, we have
Proof. With the same argument as before, we can writeFrom Lemma 7, we obtainAccording to the properties of function , we have thatFrom the above inequality and equality (25), we obtain our result.
Similarly to Theorem 4 and by using Lemma 8, we can reach the following result.
Theorem 5. Let be a nonempty and nonsingular graph with vertices. Then,
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 byfor 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. Since is increasing on and since (note that ), hence, we have , 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 and , 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. Sinceby using the properties of the function, we can write , that is, the bound in (20) is better than the bound in (14).
Data Availability
The data involved in the examples of our study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
J. Rodríguez was supported by MINEDUC-UA Project, code ANT-1899, and funded by the Initiation Program in Research-Universidad de Antofagasta, INI-19-06, and Programa Regional MATHAMSUD MATH2020003.