Minimizing the Least Eigenvalue of Bicyclic Graphs with Fixed Diameter

Abstract

Let $\mathcal{B}(n,d)$ be the set of bicyclic graphs with both $n$ vertices and diameter $d$, and let ${\theta }^{\ast }$ consist of three paths $u_0{w}_1{v}_0$, $u_0{w}_2{v}_0$, and $u_0{w}_3{v}_0$. For four nonnegative integers $n,d,k,j$ satisfying $n\ge d+3$, $d=k+j+2$, we let $B(n,d;k,j)$ denote the bicyclic graph obtained from ${\theta }^{\ast }$ by attaching a path of length $k$ to $u_0$, attaching a path of length $j$ to vertex $v_0$ and $n-d-3$ pendant edges to $w_0$, and let $\mathcal{B}(n,d;k,j)=\{B(n,d;k,j)\mid k+j\ge 1\}$. In this paper, the extremal graphs with the minimal least eigenvalue among all graphs in $\mathcal{B}(n,d;k,j)$ are well characterized, and some structural characterizations about the extremal graphs with the minimal least eigenvalue among all graphs in $\mathcal{B}(n,d)$ are presented as well.