The Signless Laplacian Spectral Radius of Tricyclic Graphs with a Given Girth

Abstract:

      A tricyclic graph $G=(V(G),E(G))$ is a connected and simple graph such that $|E(G)|=|V(G)|+2$. Let $\mathscr{T}_n^g$ be the set of all tricyclic graphs on $n$ vertices with girth $g$. In this paper, we will show that there exists the unique graph which has the largest signless Laplacian spectral radius among all tricyclic graphs with girth $g$ containing exactly three (resp., four) cycles. And at the same time, we also give an upper bound of the signless Laplacian spectral radius and the extremal graph having the largest signless Laplacian spectral radius in $\mathscr{T}_n^g$, where $g$ is even.