Huiqing Liu

Huiqing Liu 1, Mei Lu 2, and Feng Tian 3
1 Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, P. R. China
2 Department of Mathematical Sciences, Tsinghua University, Beijing 100084, P. R. China
3 School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, P. R. China

In this paper, we study the spectral radius of unicyclic graphs with $n$ vertices and diameter $d$. We determined graphs with the largest spectral radius among all the unicyclic graphs with $n$ vertices and diameter $d$. Moreover, if $4\le d\le n-3,~d\equiv 0(\mbox{mod}~2)$, then the second largest special radius of unicyclic graphs with $n$ vertices and diameter $d$ are characterized.

Let $\triangle_n^d$ be a graph of order $n$ obtained from a triangle by attaching $n-d-2$ pendant edges and a path of length $\lfloor\frac{d}{2}\rfloor$ at one vertex of triangle, and a path of length $\lceil\frac{d}{2}\rceil-1$ to another vertex of triangle, respectively.

Let $\nabla_n^d$ be a graph of order $n$ obtained from a triangle by attaching $n-d-2$ pendant edges and a path of length $\lceil\frac{d}{2}\rceil-1$ at one vertex triangle, and a path of length $\lfloor\frac{d}{2}\rfloor$ to another vertex of triangle, respectively.

Theorem 1. Let $G$ be a graph in $\mathscr U$ $_n^d,~d\ge 1$. Then $$\rho(G)\le \rho(\triangle_n^d),$$ and equality holds if and only if $G\cong \triangle_n^d$.

Theorem 2. Let $G\in\mathscr{U}_n^d\setminus\{\triangle_n^d\}$. Suppose that $d\equiv 0(\mbox{mod}~2)$ and $4\le d\le n-3$. Then $$\rho(G)\le \rho(\nabla_n^d),$$ and equality holds if and only if $G\cong \nabla_n^d$.