Improved bound of graph energy in terms of vertex cover number

TL;DR

This paper establishes a new lower bound for the energy of a graph based on its vertex cover number, improving previous bounds for various classes of graphs.

Contribution

The paper introduces a tighter lower bound for graph energy in terms of vertex cover number for multiple graph classes.

Findings

Proves $\, \, \, \\mathcal{E}(G) \\geq 2\tau$ for several graph classes.

Improves previous bound $\, \, \, \\mathcal{E}(G) \\geq 2\tau - 2c$.

Enhances understanding of the relationship between graph energy and vertex cover number.

Abstract

Let $ G $ be a simple graph with the vertex cover number $ \tau $. The energy $ \mathcal{E}(G) $ of $ G $ is the sum of the absolute values of all the adjacency eigenvalues of $ G $. In this article, we establish $ \mathcal{E}(G)\geq 2\tau $ for several classes of graphs. The result significantly improves the known result $ \mathcal{E}(G)\geq 2\tau-2c$ for many classes of graphs, where $ c $ is the number of odd cycles.