Abstract
In this study, we investigate the Laplacian degree product spectrum and corresponding energy of four families of graphs, namely, complete graphs, complete bipartite graphs, friendship graphs, and corona products of 3 and 4 cycles with a null graph.
1. Introduction
The graph energy was firstly introduced by Ivan Gutman in 1978 [1]. His idea was motivated by the well-known Hückel molecular orbital theory by Erich Hückel in 1930s, which permits pharmacologists to imprecise energies associated -electron orbital of molecules called conjugated hydrocarbons [2]. The spectrum and the energy of a graph have significant applications and connections in the branches of Mathematics, such as linear algebra and combinatorial optimization fields which have lot to do with graph spectrum and energy. The combinatorial and graph theoretical approaches have strong bonding to solve real-life problems. Many results and methods from the spectral graph theory can be applied for the practicalities and evolution of matrix theory [3]. An ordered pair , called a graph with vertex set of , is denoted by and its edge set by . Two vertices are adjacent if they make an edge in , and we denote it by . The number of edges incident to a vertex of is the degree of , and it is denoted by [4, 5]. The adjacency matrix of , of order denoted by , is a square symmetric matrix of order whose th element can be found as [3]
For energy and spectrum of graph , let be the adjacency matrix, the summation of absolute values of its eigenvalues compose energy of graph and these eigenvalues related with their multiplicities forms the spectrum of graph [4], i.e.,andwhere are the multiplicities of the eigenvalues of . In [6], the degree product adjacency matrix, for a simple connected graph having vertices say , is a real symmetric matrix, denoted by ,1 with
The Laplacian degree product adjacency matrix of is defined aswhere is the degree matrix of having diagonal entries as the degree of each vertex and all other entries are zero. The spectrum (1) and energy (2) obtained correspond to the eigenvalues of and are called the Laplacian degree product adjacency spectrum and energy, and , respectively [7], as the degree sum concept was conceived earlier in [8].
2. Main Results
In this module, we study the Laplacian degree product adjacency spectrum and energy of some well-known families of graphs, such as complete graphs, complete bipartite graphs, friendship graphs, and corona products of 3 and 4 cycles with null graph. We also evaluate the correct spectrum and the energy of degree product adjacency matrix of the corona product of 4 cycle with null graphs (thorny 4-cycle ring), which was found incorrect in [6].
2.1. Complete Graphs
Let be the vertex set of ; then, the following result provides the Laplacian degree product adjacency spectrum and energy of .
Theorem 1. For , let be a complete graph. Then,and Laplacian degree product adjacency energy of is -times the size of .
Proof. First of all note that , for each . Accordingly, we have the following Laplacian degree product adjacency matrix:Eigenvalues of areThese eigenvalues provide the required spectrum. Moreover, by (3), we haveSince the size of is , so the result is proved.
2.2. Complete Bipartite Graphs
Let a complete bipartite graph with vertex sets and be as partitions. The order and the size of graph are and , respectively. Then, the Laplacian degree product adjacency spectrum and energy of can be obtained from the following result.
Theorem 2. For , a complete bipartite graph , thenMoreover,
Proof. Note that , for each and , for each . Accordingly, the Laplacian degree product adjacency matrix of isNext, we have four cases to discuss. Case I ( with ): eigenvalues of are The required spectrum can be obtained by these eigenvalues. Furthermore, by (3), we have Case II ( with ): we get the following eigenvalues of : The required spectrum can be obtained by these eigenvalues. Moreover, by (3), we have Case III (): eigenvalues of are as follows: These eigenvalues provide the required spectrum. Furthermore, by (3), we have Case IV ( and ): we get eigenvalues of as follows: These eigenvalues provide the required spectrum. Using (3), we have the following energy of :It completes the proof.
2.3. Friendship Graphs
A friendship graph has vertices, and it can be assembled by connecting clones of the cycle with a common vertex. Let the vertex set of th copy of be , where . Let the common vertex be . Then, the vertex set of is
Theorem 3. For , let a friendship graph ; then,
Proof. In , and for . Then, the Laplacian degree product adjacency matrix is as follows:The eigenvalues of Laplacian degree product adjacency matrix of areThe required spectrum can be obtained by these eigenvalues. These eigenvalues provide the following energy:
2.4. Corona Products of 3 and 4 Cycles with Null Graphs
The corona product of graphs and is expressed as . It can be made by drawing one copy of and copies of and connecting the th vertex of with each vertex of th copy of [9–11]. Let be an -cycle with vertices and be a null graph . Then, the vertex set of iswhere the set is the vertex set of th copy of in . In this portion, we evaluate the Laplacian degree product spectrum and energy of for and 4.
Theorem 4. For , let the corona product be ; then,
Proof. Note that , for each , and , for each and . For the convenience, we let . Then, the Laplacian degree product adjacency matrix of isThe eigenvalues obtained from the above matrix of areThen, the required spectrum can be obtained by these eigenvalues. Also, by (3), we have
Theorem 5. For , let the corona product be ; then, is
Proof. Note that , for each, and , for each and . For the convenience, we let . Then, the Laplacian degree product adjacency matrix of is asThe eigenvalues of Laplacian degree product adjacency matrix of areThen, the required spectrum can be obtained by these eigenvalues. Furthermore, by (3), we have
3. Appendix
In [6], Mirajkar and Doddamani considered the corona product for (also called thorny cycle rings ) and investigated its energy and spectrum on the base of degree product adjacency matrix. During computations of our results on , the eigenvalues investigated in [6] were found incorrect. In this section, we provide the correct energy and spectrum of . First of all, note that the degree product adjacency matrix of is
Mirajkar and Doddamani’s investigated eigenvalues are
Whereas, the correct eigenvalues of are
Accordingly, we get the following energy and spectrum of in the corrected form:
4. Accomplishment Remarks
In this study, we construct the general formulas for spectrums and energies of four different families of graphs, by using Laplacian degree product adjacency matrix. We also obtained faultless and correct eigenvalues of , which were defined in [6].
Data Availability
All kinds of data and materials, used to compute the results, are provided in Section 1.
Conflicts of Interest
The authors declare no conflicts of interest.
Authors’ Contributions
Asim Khurshid carried out computation and wrote the initial draft; Muhammad Salman supervised the study, administrated the project, analysed data, and developed the methodology; Masood ur Rehman: wrote and reviewed the manuscript, carried out formal analysis, and edited the study; Mohammad Tariq Rahim conceptualized the study and visualized the study.