On the Laplacian Spectral Radius of Unicyclic Graphs with Fixed Diameter

Abstract

The set of unicyclic graphs with $n$ vertices and diameter $d$ is denoted by ${\mathcal{U}}_{n,d}$. For $3\le i\le d$, let $P_{n-d-1}(i)$ be the graph obtained from path $P_{d+1}:v_1{v}_2\dots v_{d+1}$ by adding $n-d-1$ pendant edges at $v_i$, and $U_{n-d-2}(i)$ be the graph obtained from $P_{n-d-1}(i)$ by joining $v_{i-2}$ and a pendant neighbor of $v_i$. In this paper, we determine all unicyclic graphs in ${\mathcal{U}}_{n,d}$ whose largest Laplacian eigenvalue is greater than $n-d+2$. For $n-d\ge 6$ and $G\in {\mathcal{U}}_{n,d}$, we prove further that the largest Laplacian eigenvalue $\mu (G)\le max\{\lambda (U_{n,d-2}(i))\mid 3\le i\le d\}$, and conjecture that ${\mathcal{U}}_{n,d}.$ is the unique graph which has the greatest value of the greatest Laplacian eigenvalue in ${\mathcal{U}}_{n,d}$. We also prove that the conjecture is true for $3\le d\le 6$.