A (forgotten) upper bound for the spectral radius of a graph

TL;DR

This paper revisits a historical upper bound for the spectral radius of a graph, demonstrating its equivalence to a recent bound, and applies it to compare clique number bounds, propose new bounds for the signless Laplacian spectral radius, and extend a result for r-partite graphs.

Contribution

It uncovers the equivalence of a forgotten 1983 spectral radius bound with a recent one and applies it to derive new bounds for graph invariants.

Findings

The 1983 upper bound is equivalent to Liu and Weng's bound.

New sharp upper bounds for the signless Laplacian spectral radius are proposed.

A lower bound for generalized r-partite graphs extending Erdős's result is proved.

Abstract

The best degree-based upper bound for the spectral radius is due to Liu and Weng. This paper begins by demonstrating that a (forgotten) upper bound for the spectral radius dating from 1983 is equivalent to their much more recent bound. This bound is then used to compare lower bounds for the clique number. A series of sharp upper bounds for the signless Laplacian spectral radius is then proposed as another application. Finally a new lower bound for generalised r-partite graphs is proved, by extending a result due to Erdos.