Bulletin, Classe des Sciences Mathématiques et Naturelles, Sciences mathématiques naturelles / sciences mathematiques Vol. CXXXI, No. 30, pp. 93–99 (2005) 

On the spectral radius of bicyclic graphsM. Petrovic, I. Gutman and ShuGuang GuoFaculty of Science, University of Kragujevac, P. O. Box 60, 34000 Kragujevac, Serbia and MontenegroDepartment of Mathematics, Yancheng Teachers College, Yancheng 224002, Jiangsu, P. R. China 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 5vertex graph, obtained by identifying a vertex of $K_3$ with a vertex of $K_3'$ . Let $B_n^{\ast\ast}(0)$ be the 4vertex 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 Full text of the article: (for faster download, first choose a mirror)
Electronic fulltext finalized on: 21 Nov 2005. This page was last modified: 20 Jun 2011.
© 2005 Mathematical Institute of the Serbian Academy of Science and Arts
