On the spectral radius of bicyclic graphs


M. PetroviŠ, I. Gutman, Shu-Guang Guo


Let $K_3$ and $K_3'$ be two complete graphs of order 3 with disjoint vertex sets. Let $B_n^{\ast}(0)$ be the 5-vertex graph, obtained by identifying a vertex of $K_3$ with a vertex of $K_3'$ . Let $B_n^{\ast\ast}(0)$ be the 4-vertex graph, obtained by identifying two vertices of $K_3$ each with a vertex of $K_3'$ . Let $B_n^{\ast}(k)$ be graph of order $n$ , obtained by attaching $k$ paths of almost equal length to the vertex of degree 4 of $B_n^{\ast}(0)$ . Let $B_n^{\ast\ast}(k)$ be the graph of order $n$ , obtained by attaching $k$ paths of almost equal length to a vertex of degree 3 of $B_n^{\ast\ast}(0)$ . Let ${\cal B}_n(k)$ 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 ${\cal B}_n(k)$ , either $B_n^{\ast}(k)$ or $B_n^{\ast\ast}(k)$ have the greatest spectral radius. We now show that for $k \geq 1$ and $n \geq k+5$ , among the elements of ${\cal B}_n(k)$ , the graph $B_n^{\ast}(k)$ has the greatest spectral radius.