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

Let \(\dot{G}\) $\dot{G}$ be a signed graph and \(A(\dot{G})\) $A(\dot{G})$ be its adjacency matrix. The eigenvalues of \(\dot{G}\) $\dot{G}$ are actually the eigenvalues of \(A(\dot{G})\) $A(\dot{G})$ , and the girth of \(\dot{G}\) $\dot{G}$ is the length of a shortest cycle in \(\dot{G}\) $\dot{G}$ . We use \(\mathscr {B}(n,g)\) $\mathcal{B}(n,g)$ 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 \(\mathscr {B}(n,g)\) $\mathcal{B}(n,g)$ and accordingly determine the extremal signed graph which achieves the minimal least eigenvalue.