On the Spectral Radius of Bicyclic Graphs

A b s t r a c t. Let K3 and K ′ 3 be two complete graphs of order 3 with disjoint vertex sets. Let B∗ n(0) be the 5-vertex graph, obtained by identifying a vertex of K3 with a vertex of K ′ 3 . Let B∗∗ n (0) be the 4-vertex graph, obtained by identifying two vertices of K3 each with a vertex of K ′ 3 . Let B∗ n(k) be graph of order n , obtained by attaching k paths of almost equal length to the vertex of degree 4 of B∗ n(0) . Let B∗∗ n (k) be the graph of order n , obtained by attaching k paths of almost equal length to a vertex of degree 3 of B∗∗ n (0) . Let Bn(k) be the set of all connected bicyclic graphs of order n , possessing k pendent vertices. One of the authors recently proved that among the elements of Bn(k) , either B∗ n(k) or B∗∗ n (k) have the greatest spectral radius. We now show that for k ≥ 1 and n ≥ k + 5 , among the elements of Bn(k) , the graph B∗ n(k) has the greatest spectral radius. AMS Mathematics Subject Classification (2000): 05C50, 05C35