Abstract

Let G = (V (G), E (G)) be a unicyclic simple undirected graph with largest vertex degree Δ. Let C r be the unique cycle of G. The graph G - E (C r) is a forest of r rooted trees T 1, T 2, ..., T r with root vertices v 1, v 2, ..., v r, respectively. Letk (G) = under(max, 1 ≤ i ≤ r) {max {dist (v i, u) : u ∈ V (T i)}} + 1,where dist (v, u) is the distance from v to u. Let μ 1 (G) and λ 1 (G) be the spectral radius of the Laplacian matrix and adjacency matrix of G, respectively. We prove thatμ 1 (G) < Δ + 2 sqrt(Δ - 1) cos frac(π, 2 k (G) + 1),whenever Δ > 2 andλ 1 (G) < 2 sqrt(Δ - 1) cos frac(π, 2 k (G) + 1),whenever Δ ≥ 4 or whenever Δ = 3 and k (G) ≥ 4. © 2007 Elsevier Inc. All rights reserved.