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

DOI：10.3770/j.issn:2095-2651.2014.04.001

 作者 单位 乔露 西北工业大学理学院应用数学系, 陕西 西安 710072 王力工 西北工业大学理学院应用数学系, 陕西 西安 710072

一个三圈图\$G=(V(G),E(G))\$是一个满足\$|E(G)|=|V(G)|+2\$的简单连通图.记\$\mathscr{T}_n^g\$是所有顶点数是\$n\$围长为\$g\$的三圈图所组成的集合.在这篇文章中, 我们将证明, 在给定围长\$g\$且恰好含有三个圈(或四个圈)的三圈图所组成的集合中存在唯一的三圈图, 使得它在这一集合中具有最大的无符号拉普拉斯谱半径. 同时, 我们也给出了当 \$g\$是偶数时,在\$\mathscr{T}_n^g\$中的无符号拉普拉斯谱半径的上界以及具有最大无符号拉普拉斯谱半径的极图.

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.