New upper bounds on the spectral radius of unicyclic graphs

Abstract

G = (V (G),E(G)) be a unicyclic simple undirected graph with largest vertex degree . Let Cr be the unique cycle of G. The graph G − E(Cr ) is a forest of r rooted treesT1,T2, . . .,Tr with root vertices v1, v2, . . ., vr , respectively. Let k(G) = max 1 i r {max{dist(vi, u) : u ∈ V (Ti )}} + 1, where dist(v, u) is the distance from v to u. Let μ1(G) and λ1(G) be the spectral radius of the Laplacian matrix and adjacency matrix of G, respectively. We prove that μ1(G) < + 2

− 1 cos π 2k(G) + 1 , whenever > 2 and λ1(G) < 2

− 1 cos π 2k(G) + 1 , whenever 4 or whenever = 3 and k(G) 4. © 2007 Elsevier Inc. All rights reserved. AMS classification: 5C50; 15A48; 05C05