On the spectral radius of bicyclic graphs

M. Petrovic, I. Gutman and Shu-Guang Guo

Abstract: 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.

Keywords: spectrum (of graph), spectral radius (of graph), bicyclic graphs, extremal graphs

Classification (MSC2000): 05C50, 05C35

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