Abstract
The aggregate of the absolute values of the graph eigenvalues is called the energy of a graph. It is used to approximate the total _-electron energy of molecules. Thus, finding a new mechanism to calculate the total energy of some graphs is a challenge; it has received a lot of research attention. We study the eigenvalues of a complete tripartite graph Ti,i,n−2i , for n _ 4, based on the adjacency, Laplacian, and signless Laplacian matrices. In terms of the degree sequence, the extreme eigenvalues of the irregular graphs energy are found to characterize the component with the maximum energy. The chemical HMO approach is particularly successful in the case of the total _-electron energy. We showed that some chemical components are equienergetic with the tripartite graph. This discovering helps easily to derive the HMO for most of these components despite their different structures.
Highlights
The general formula of eigenvalues of Ti,i,n−2i based on adjacency matrix A is given by
The general formula of eigenvalues of Ti,i,n−2i based on Laplacian matrix L is given by
The general formula of eigenvalues of Ti,i,n−2i based on signless Laplacian matrix S is given by i)(2i−2), (2i)(n−2i)−1, B
Summary
Properties of Ti,i,n−2i: 1. The number of cycle C with length 3 in Ti,i,n−2i is equal i(n − 2i). The general formula of eigenvalues of Ti,i,n−2i based on adjacency matrix A is given by. The general formula of eigenvalues of Ti,i,n−2i based on Laplacian matrix L is given by. To simplify, based on Laplacian matrix, some properties of the spectrum of Ti,i,n−2i are given as follows. The proofs of these three properties can be calculated from the general spectral formula of Ti,i,n−2i based on L(Ti,i,n−2i). The general formula of eigenvalues of Ti,i,n−2i based on signless Laplacian matrix S is given by. For Ti,i,n−2i, the characteristic polynomial is obtained by computing the determinant of |S (Ti,i,n−2i) − λI| which is equal to zero and the signless matrix of Ti,i,n−2i is shown in (5.2) as. T4,4,n−8 1, 56, 5.6277, 11.3723 2, 8, 66, 4.8769, 13.1231 3, 76, 82, 4.3765, 11.3723
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