On the spectral radius of graphs with given maximum degree and girth

Abstract

We prove upper bounds for the spectral radius ρ(G) of an n-vertex graph with given maximum degree and girth at least 2ℓ+1. This extends the previous result of [9] regarding graphs with girth at least five. When ℓ=3 or |V(G)| is relatively small compared with the maximum degree, our upper bounds are sharp. In addition, for a tree T, we provide an upper bound for the spectral radius of an n-vertex T-free graph with given maximum degree. This bound is also sharp for a certain class of trees.