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 \(k\).