Energy of a graph and Randi\'c index of subgraphs

TL;DR

This paper introduces a new inequality linking the energy of a graph to the Randić index of its subgraphs, generalizing existing bounds and providing new insights into graph spectral properties.

Contribution

It establishes a novel inequality connecting graph energy and Randić index of subgraphs, broadening the scope of spectral graph theory results.

Findings

Proves that the energy of a graph is at least twice the Randić index of any subgraph.

Generalizes known inequalities involving graph energy, Randić index, and matching number.

Provides additional inequalities as applications of the main result.

Abstract

We give a new inequality between the energy of a graph and a weighted sum over the edges of the graph. Using this inequality we prove that $\mathcal{E}(G)\geq 2R(H)$, where $ \mathcal{E}(G)$ is the energy of a graph $G$ and $R(H)$ is the Randi\'c index of any subgraph of $G$ (not necessarily induced). In particular, this generalizes well-known inequalities $\mathcal{E}(G)\geq 2R(G)$ and $\mathcal{E}(G)\geq 2\mu(G)$ where $\mu(G)$ is the matching number. We give other inequalities as applications to this result.