Minimizing the Least Eigenvalue of Bicyclic Graphs with $k$ Pendant Vertices

Abstract

Let $\mathcal{B}(n,k)$ be the set of bicyclic graphs with $n$ vertices and $k$ pendant vertices. In this paper, we determine the unique graph with minimal least eigenvalue among all graphs in $\mathcal{B}(n,k)$. This extremal graph is the same as that on the Laplacian spectral radius as done by Ji-Ming Guo(The Laplacian spectral radius of bicyclic graphsmwith $n$ vertices and $k$ pendant vertices, Science China Mathematics, $53(8)(2010)2135-2142]$. Moreover, the minimal least eigenvalue is a decreasing function on $k$.