Abstract
The degree sum exponent distance matrix M(G)of a graph G is a square matrix whose (i,j)-th entry is (di+dj)^ d(ij) whenever i not equal to j, otherwise it is zero, where di is the degree of i-th vertex of G and d(ij)=d(vi,vj) is distance between vi and vj. In this paper, we define degree sum exponent distance energy E(G) as sum of absolute eigenvalues of M(G). Also, we obtain some bounds on the degree sum exponent distance energy of some graphs and deduce direct expressions for some graphs.
Highlights
The concept of graph energy was introduced by I.Gutman in 1978[1] having direct correlation with the total π-electron energy of a molecule in the quantum chemistry as calculated with the Huckel molecular orbital method
Several results on energy related with degree of a vertex and distance in a graph were studied such as distance energy [5],degree sum energy of some graphs [6], degree square sum polynomial of some graphs [8], degree sum energy [9],a survey on energy of graphs [7], complementary distance energy[10], degree sum distance energy [11], degree product distance energy[12],degree exponent energy[13] and degree exponent sum energy[16]
In order to upgrade, we introduce concept of degree sum exponent distance energy of connected graph which is slight generalization of degree sum energy since if exponent is made one, it coincides with degree sum energy
Summary
The concept of graph energy was introduced by I.Gutman in 1978[1] having direct correlation with the total π-electron energy of a molecule in the quantum chemistry as calculated with the Huckel molecular orbital method. For every pair of vertices in a connected graph there are, degree associated each one of them and in addition there is distance between them In order to upgrade, we introduce concept of degree sum exponent distance energy of connected graph which is slight generalization of degree sum energy since if exponent is made one, it coincides with degree sum energy. The purpose of this paper is to compute the characteristic polynomial, eigenvalues and energy of the new matrix associated with graph, called degree sum exponent distance matrix, and compute bounds for degree sum exponent distance energy and obtain expressions for some standard graphs
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