Bipartite unicyclic graphs with a unique perfect matching having the smallest positive eigenvalue equal to

The smallest positive eigenvalue $ \tau (G) $ tau(G) of a simple graph G is the smallest positive eigenvalue of its adjacency matrix $ A(G) $ A(G). In [F. J. Zhang and A. Chang, Acyclic molecules with greatest HOMO-LUMO separation, Discrete Applied Mathematics, 98:165-171, (1999).], the authors characterized all nonsingular trees with tau equal to $ \sqrt {2}-1 $ 2-1. We consider the same problem for bipartite unicyclic graphs with a unique perfect matching. Let $ \mathcal {U} $ U be the class of all connected bipartite unicyclic graphs with a unique perfect matching. In this article, we characterize all graphs U in $ \mathcal {U} $ U with the property that $ \tau (U)=\sqrt {2}-1 $ tau(U)=2-1. Further, we show that the largest limit point of the smallest positive eigenvalues of graphs in $ \mathcal {U} $ U is $ \sqrt {2}-1 $ 2-1, whereas the smallest limit point is 0.