On the spectral radius of bicyclic graphs

Abstract

Let K3 and K'3 be two complete graphs of order 3 with disjoint vertex sets Let
be the 5-vertex graph, obtained by identifying a vertex of K3 with a vertex of K'3. Let
be the 4-vertex graph, obtained by identifying two vertices of K3 each with a vertex of K'3. Let
be graph of order n, obtained by attaching k paths of almost equal length to the vertex of degree 4 of
. Let
be the graph of order n obtained by attaching k paths of almost equal length to a vertex of degree 3 of
. Let
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
, either
or
have the greatest spectral radius. We now show that for k ≥ 1 and n ≥ k + 5, among the elements of
, the graph
has the greatest spectral radius.