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

Abstract

We prove upper bounds for the spectral radius rho ( G ) of an nvertex graph with given maximum degree and girth at least 2 $ +1. This extends the previous result of [9] regarding graphs with girth at least five. When $ = 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 nvertex T - free graph with given maximum degree. This bound is also sharp for a certain class of trees. (c) 2024 Elsevier Inc. All rights reserved.