On the smallest positive eigenvalue of bipartite graphs with a unique perfect matching

Abstract

Let G be a simple graph with the adjacency matrix A(G), and let ?(G) denote the smallest positive eigenvalue of A(G). Let Gn be the class of all connected bipartite graphs on n=2k vertices with a unique perfect matching. In this article, we characterize the graphs G in Gn such that ?(G) does not exceed [Formula presented]. Using the above characterization, we obtain the unique graphs in Gn with the maximum and the second maximum ?, respectively. Further, we prove that the largest and the second largest limit points of the smallest positive eigenvalues of bipartite graphs with a unique perfect matching are [Formula presented] and the reciprocal of [Formula presented], respectively, where ?3 is the largest root of x3?x?1. � 2024 Elsevier B.V.