New upper bounds on the spectral radius of unicyclic graphs

Abstract: Let be a unicyclic simple undirected graph with largest vertex degree Δ. Let be the unique cycle of . The graph is a forest of r rooted trees with root vertices , respectively. Letwhere is the distance from v to u. Let and be the spectral radius of the Laplacian matrix and adjacency matrix of , respectively. We prove thatwhenever andwhenever or whenever and . [Copyright &y& Elsevier]