Arithmetic–geometric energy of specific graphs

Abstract

Let [Formula: see text] be a simple graph of order [Formula: see text] with vertex set [Formula: see text] and edge set [Formula: see text]. The arithmetic–geometric matrix [Formula: see text] of [Formula: see text] is a matrix of order [Formula: see text] defined by [Formula: see text] if [Formula: see text] and 0 otherwise, where [Formula: see text] stands for the degree of the vertex [Formula: see text] in [Formula: see text]. The arithmetic–geometric characteristic polynomial of [Formula: see text] is the characteristic polynomial of [Formula: see text]. The arithmetic–geometric energy [Formula: see text] of [Formula: see text] is the sum of absolute values of all eigenvalues of [Formula: see text]. In this paper, we obtain the arithmetic–geometric characteristic polynomial and arithmetic–geometric energy of some specific graphs. In addition, we also consider the arithmetic–geometric characteristic polynomial and arithmetic–geometric energy change of these graphs when one edge is deleted.

On Some Topological Indices Defined via the Modified Sombor Matrix

“…The following facts seem to be interesting to note about some particular weighted adjacency matrices: The energy of a graph cannot be an odd integer (Bapat and Pati [ 36 ]). The arithmetic–geometric energy of a graph can be any integer greater than one (Zheng, Tian and Cui [ 37 ]). The modified Sombor energy of every regular complete multipartite is .…”

“…The arithmetic–geometric energy of a graph can be any integer greater than one (Zheng, Tian and Cui [ 37 ]).…”

“…Some notable points regarding some weighted adjacency matrices are below: The classical graph energy cannot be an odd integer (see [ 36 ]). The arithmetic–geometric energy can be any positive integer greater than one (see [ 37 ]). The modified Sombor energy of every regular complete multipartite is constant and equals (see Corollary 1).…”