Choice Number and Energy of Graphs

TL;DR

This paper establishes new lower bounds on the energy of a graph based on its chromatic and choice numbers, linking spectral properties to coloring parameters.

Contribution

It introduces novel inequalities relating graph energy to chromatic and choice numbers, expanding understanding of spectral graph theory.

Findings

E(G) >= 2(n - χ(Ḡ)) for all graphs G

E(G) >= 2ch(G) for most graphs

Identifies specific families where E(G) >= 2ch(G) does not hold

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. It is proved that E(G)>= 2(n-\chi(\bar{G}))>= 2(ch(G)-1) for every graph G of order n, and that E(G)>= 2ch(G) for all graphs G except for those in a few specified families, where \bar{G}, \chi(G), and ch(G) are the complement, the chromatic number, and the choice number of G, respectively.