Energy of a graph and Randic index

TL;DR

This paper establishes a lower bound for the energy of any graph in terms of its Randic index, with equality characterized by the graph being a union of complete bipartite graphs.

Contribution

It proves a new inequality relating graph energy and Randic index, and characterizes the extremal graphs where equality holds.

Findings

Graph energy is at least twice the Randic index for any graph.

Equality occurs if and only if the graph is a union of complete bipartite graphs.

Abstract

We prove that, for any graph $G$, its graph energy is at least twice the Randic index. We show that equality holds if and only if $G$ is the union of complete bipartite graphs.