New bounds of extended energy of a graph

TL;DR

This paper introduces the concept of extended energy of a graph based on an extended adjacency matrix, and establishes new bounds that improve upon existing ones, including Nordhaus-Gaddum-type bounds.

Contribution

It defines extended vertex energy and derives new upper and lower bounds for the extended energy of graphs, improving previous bounds and providing new inequalities.

Findings

New bounds involving graph order, size, and degrees

Improved upper bounds over existing results

Nordhaus-Gaddum-type bounds for extended energy

Abstract

The extended adjacency matrix of a graph with $n$ vertices is a real symmetric matrix of order $n\times n$ whose $(i,j)$-th entry is the average of the ratio of the degree of the vertex $i$ to that of the vertex $j$ and its reciprocal when $i,j$ are adjacent and zero otherwise. The aggregate of absolute eigenvalues of the extended adjacency matrix is termed the extended energy. In this paper, the concept of extended vertex energy is introduced, and some bounds of extended vertex energy are obtained. From there, we establish some new upper bounds of the extended energy of a graph involving order, size, largest, and smallest degree. We show that those are improvements of some existing bounds. Through direct manipulation, we have also established some more upper and lower bounds of extended energy, which are either better or incomparable with the existing bounds. Finally, some improved bounds of Nordhaus-Gaddum-type are found.