On the least eigenvalues of unbalanced signed bicyclic graphs with given girth

Abstract

Let \(\dot{G}\) be a signed graph and \(A(\dot{G})\) be its adjacency matrix. The eigenvalues of \(\dot{G}\) are actually the eigenvalues of \(A(\dot{G})\), and the girth of \(\dot{G}\) is the length of a shortest cycle in \(\dot{G}\). We use \(\mathscr {B}(n,g)\) to denote the set of unbalanced signed bicyclic graphs on n vertices with girth g. In this paper, we focus on the least eigenvalues of signed graphs in \(\mathscr {B}(n,g)\) and accordingly determine the extremal signed graph which achieves the minimal least eigenvalue.