The Signless Laplacian Spectral Radius of Unicyclic and Bicyclic Graphs with a Given Girth

摘要

Let $mathcal{U}(n,g)$ and $mathcal{B}(n,g)$ be the set of unicyclic graphs and bicyclic graphs on $n$ vertices with girth $g$, respectively. Let $mathcal{B}_{1}(n,g)$ be the subclass of $mathcal{B}(n,g)$ consisting of all bicyclic graphs with two edge-disjoint cycles and $mathcal{B}_{2}(n,g)=mathcal{B}(n,g)ackslashmathcal{B}_{1}(n,g)$. This paper determines the unique graph with the maximal signless Laplacian spectral radius among all graphs in $mathcal{U}(n,g)$ and $mathcal{B}(n,g)$, respectively. Furthermore, an upper bound of the signless Laplacian spectral radius and the extremal graph for $mathcal{B}(n,g)$ are also given.