(Generalized) Incidence and Laplacian-Like Energies

Abstract

In this study, for graph with connected components (also for connected nonbipartite and connected bipartite graphs) and a real number , we found generalized and improved bounds for the sum of -th powers of Laplacian and signless Laplacian eigenvalues of . Consequently, we also generalized and improved results on incidence energy and Laplacian energy-like invariant .

1. Introduction

Let denote a finite, simple, and undirected graph of order . The edge and vertex sets of are denoted by and , respectively. If the vertex is neighbour to , then write . The degree of the vertex , symbolized by , is the number of vertices adjacent to .

The adjacency matrix and the degree matrix of graph are denoted by and respectively. Let be the eigenvalues of the Laplacian matrix of where [1, 2]. Let be the eigenvalues of the signless Laplacian matrix of where [3]. Since the matrices , , and are real and symmetric matrices, thus they have real eigenvalues. So, we can write their eigenvalues such that , , and respectively. and are semidefinite matrices, according to the Geršgorin disc theorem. From here, all eigenvalues of Laplacian and signless Laplacian matrices of are non-negative integers. In [3], it has been found that for a connected nonbipartite graph . Additionally, is a bipartite graph if and only if .

The link between the eigenvalues of a graph and the molecular orbital energy levels of electrons in conjugated hydrocarbons is the most crucial chemical application of graph theory. The total electron energy in conjugated hydrocarbons is calculated by the sum of absolute values of the eigenvalues corresponding to the molecular graph which has a maximum of four degree generally for the Hüchkel molecular orbital approximation. The energy of given by Gutman in [4] is as follows:

Nowadays, there is a lot of study on graph energy, as can be seen from the recent papers [5].

The square roots of the eigenvalues of the matrix are known as the singular values of some matrix and its transpose . Recently, in [2], Nikiforov introduced and explored the notion of graph energy. He defined the energy of a graph to be the sum of singular values of any matrix . Clearly,

Assume that represents the vertex-edge incidence matrix of the graph . Then, for having vertex set and edge set , the entry of is if is not incident with and if is incident with . Jooyandeh et al. [6] introduced the notion of incidence energy of a graph. Accordingly, the incidence energy of is the sum of the singular values of the incidence matrix of . The following expression is given by Gutman et al. [7]:

Some basic information on may be seen in [6, 7].

As abovementioned, one can compute the incidence energy of a graph by calculating the eigenvalues of signless Laplacian matrix of . However, the problem is much more complicated for some classes of graphs due to the computational complexity of finding eigenvalues of signless Laplacian matrix. Thus, to compute the invariant for some classes of graphs, it is crucial to find their lower and upper bounds. Zhou [8] found the upper bounds on the incidence energy in terms of the first Zagreb index. Different lower and upper bounds on have been studied by various researchers.

In [9], associated to the Laplacian eigenvalues, authors introduced the invariant called the Laplacian energy-like invariant (or Laplacian-like energy) which is defined as follows:

Firstly, it was examined in [9] that and Laplacian energy have similar characteristics. It has also been shown that it resembles to graph energy much more closely. For detailed information, see [10].

For a graph of order and a real number not equal to and in [8], the sum of the th powers of the nonzero Laplacian eigenvalues is defined as follows:

If is and , then the cases are trivial as and , where denotes the cardinality of the edge set of . It is clear that is equal to . We should note that is also equal to the Kirchhoff index of (for more detail (one can see [11, 12]). Many studies on have recently been published in the literature. For details, see [13, 14].

Similar to the definitions of , , and , Akbari et al. [15] defined the sum of the th powers of the signless Laplacian eigenvalues of as follows:and they also gave some connections between and . If is and , then the cases are trivial as and . Note that is equal to the incidence energy . We observed that Laplacian eigenvalues and signless Laplacian eigenvalues of bipartite graphs are equal [1, 3, 16]. Therefore, for bipartite graphs, and are equal, and hence, is equal to [17]. Recently, different properties, as well as different lower and upper bounds of have been established in [15, 17, 18].

Lemma 1 (see [19]). Let be nonnegative numbers. Then,

The equality among them holds if and only if .

We aim to obtain some strong bounds using the efficient inequality technique in Lemma 1 for main results. Also, we give some generalizations for , , indicence energy , and the Laplacian energy-like invariant of graphs (with connected components, connected nonbipartite, and connected bipartite).

The following main lemmas are required for our main results.

Let denote the number of spanning trees of a graph . Let be the Cartesian product of the graphs and . We define the following number for a graph .

Lemma 2 (see [20]). If is a connected bipartite graph with vertices, then . If is a connected nonbipartite graph with vertices, then .

Lemma 3 (see [21]). Let be a connected graph with vertices and maximum degree . Then, if and only if or or .

Lemma 4 (see [21]). Let be a connected graph of order . Then, if and only if .

Lemma 5 (see [3]). The spectra of and coincide if and only if the graph is bipartite.

2. Main Results

After above preliminary informations, we are ready to give our main results.

It is well known that if a graph has connected components, the spectrum of is the union of the spectra of , (and multiplicities are added). The same also holds for the Laplacian and the signless Laplacian spectrum.

Firstly, we give lower and upper bounds on and for a graph with connected components.

Theorem 6. Let be a graph of order with connected components such that of them are connected bipartite. Then,where and . Equalities occur in both bounds if and only if and , respectively.

Proof. Note that is an eigenvalue of Laplacian matrix with multiplicity . Taking , replacing by in Lemma 1, we obtain the following equation:whereSince , we have the following equation:Observe thatHence, we get the result.
From Lemma 1, the equalities hold if and only if .
It is known that is an eigenvalues of signless Laplacian matrix with multiplicity . For , the proof is similar, replacing by and taking in Lemma 1.
As a special case, if we take , we get the bounds for the and given as follows:

Corollary 7. Let be a graph of order with connected components such that of them are connected bipartite. Then,where and . Equalities hold in both bounds if and only if and , respectively.

Note that, if we take and in Theorem 6, we reach the following result.

Corollary 8. Let be a nonbipartite connected graph of order . Let and be as given in Lemma 2. Then,and

Inequalities (14) and (15) hold in both bounds if and only if and , respectively.

Taking in Corollary 7, we have the following corollary.

Corollary 9. Let be a nonbipartite connected graph of order and and be as given in Lemma 2. Then,and

Equalities (16) and (17) hold in both bounds if and only if and , respectively.

Now, we consider the bipartite graph case of the above theorem (Theorem 6). In the next corollary, we actually improved the results which were obtained in [22].

Corollary 10. Let be a connected bipartite graph with vertices. Let be as given in Lemma 2. Then,and

Equalities (18) and (19) hold in both bounds if and only if , , or , where is the maximum degree.

As it is well known in graph theory, every tree is bipartite. In addition, for a tree , and . From Corollary 10, we have the following.

Corollary 11. Let be a tree of order . Then,

Equalities hold in both bounds if and only if .

Remark 12. It is pertinent to mention here that in equations (15) and (17), for connected nonbipartite graphs, we recover the same lower bounds as in Theorem 2.6 (i) and Corollary 2.7 (i) in [22] through a different approach. For connected bipartite graphs, it can be seen that lower bounds (18) and (19) are better than lower bounds obtained in Theorem 2.6 (ii) and Corollary 2.7 (ii) in [22], respectively. Moreover, we obtained extra upper bounds for the relevant parameters and generalized them as different forms [22].

3. Accomplishment Remarks

In this paper, we have obtained new results for the graph invariants and of a simple graph with connected components (connected nonbipartite and connected bipartite), where is a real number. Also, as a result, we generalized and improved the results on incidence energy and Laplacian energy-like invariant .

Data Availability

All data and materials used to obtain the results are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.