Some upper bounds on the spectral radius of a graph

Abstract

For a graph G, the spectral radius ρ(G) of G is the largest eigenvalue of its adjacency matrix. The coalescence of two graphs H with a root v and K with a root w is obtained by identifying v and w from disjoint union of H and K. In this paper, we investigate the upper bounds of the spectral radius of the coalescence of two graphs, which generalize some results by Passbani and Salemi in 2019. Furthermore, we show that if the graph Guv⁎ is obtained from a connected graph G by contracting the internal edge uv which not contained in any triangle of G, thenρ(G)≤ρ(Guv⁎), the corresponding extremal graphs are characterized completely, and also extend a result of Hoffman and Smith. As an application, a new sharp upper bound on the spectral radius of a tree is provided.