The Laplacian Spectral Radius of a Class of Unicyclic Graphs

Abstract

Let $C(n,k)$ be the set of all unicyclic graphs with $n$ vertices and cycle length $k$. For any $U\in C(n,k)$, $U$ consists of the (unique) cycle (say $C_k$) of length $k$ and a certain number of trees attached to the vertices of $C_k$ having (in total) $n-k$ edges. If there are at most two trees attached to the vertices of $C_k$, where $k$ is even, we identify in the class of unicyclic graphs those graphs whose Laplacian spectral radii are minimal.