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

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