The laplacian spectral radius of tricyclic graphs with a given girth

Abstract

A tricyclic graph is a connected graph in which the number of edges equals the number of vertices plus two. Let _n^g be the class of all n-vertex tricyclic graphs with girth g. This paper determines the unique graph with the maximal Laplacian spectral radius among all graphs in _n^g with exactly three (resp. four) cycles. Furthermore, the upper bound of the Laplacian spectral radius and the extremal graph in _n^g are also obtained, where g is even.