Some Relations between Rank, Chromatic Number and Energy of Graphs

TL;DR

This paper explores the relationship between graph energy, rank, and chromatic number, characterizing graphs with energy equal to their adjacency matrix rank and establishing new bounds involving these parameters.

Contribution

It characterizes all graphs where energy equals rank and introduces new lower bounds for energy based on rank and chromatic number.

Findings

Graphs with energy equal to rank are fully characterized.

A lower bound for energy involving chromatic number and graph order is established.

New bounds for energy in terms of rank are provided.

Abstract

The energy of a graph $G$, denoted by $E(G)$, is defined as the sum of the absolute values of all eigenvalues of $G$. Let $G$ be a graph of order $n$ and ${\rm rank}(G)$ be the rank of the adjacency matrix of $G$. In this paper we characterize all graphs with $E(G)={\rm rank}(G)$. Among other results we show that apart from a few families of graphs, $E(G)\geq 2\max(\chi(G), n-\chi(\bar{G}))$, where $n$ is the number of vertices of $G$, $\bar{G}$ and $\chi(G)$ are the complement and the chromatic number of $G$, respectively. Moreover some new lower bounds for $E(G)$ in terms of ${\rm rank}(G)$ are given.