The energy change of the complete multipartite graph

TL;DR

This paper investigates how the energy of complete multipartite graphs changes upon the removal of an edge, especially when some parts have size one, providing a complete characterization of this energy variation.

Contribution

It extends previous results by analyzing the energy change in complete multipartite graphs when parts of size one are involved, offering a full description of the energy variation upon edge deletion.

Findings

Determines energy change for graphs with parts of size one.

Provides a complete characterization of energy variation upon edge removal.

Extends known results to broader class of multipartite graphs.

Abstract

The energy of a graph is defined as the sum of the absolute values of all eigenvalues of the graph. Akbari et al. \cite{S. Akbari} proved that for a complete multipartite graph $K_{t_1 ,\ldots,t_k}$, if $t_i\geq 2 \ (i=1,\ldots,k)$, then deleting any edge will increase the energy. A natural question is how the energy changes when $\min\{t_1 ,\ldots,t_k\}=1$. In this paper, we will answer this question and completely determine how the energy of a complete multipartite graph changes when one edge is removed.