Cubic bipartite graphs with minimum spectral gap

Abstract

Algebraic connectivity of a graph is the second smallest eigenvalue of its Laplacian matrix. Abdi, Ghorbani and Imrich, in [European J. Combin. 95 (2021) 103328], showed that the minimum algebraic connectivity of cubic graphs on 2n vertices is (1 + o(1)) π 2 2n 2 . The minimum value is attained on non-bipartite graphs. In this paper, we investigate the algebraic connectivity of cubic bipartite graphs. We prove that the minimum algebraic connectivity of cubic bipartite graphs on 2n vertices is (1 + o(1)) π 2 n 2 . Moreover, the unique cubic bipartite graph with minimum algebraic connectivity is determined. Based on the relation between the algebraic connectivity and spectral gap of regular graphs, the cubic bipartite graph with minimum spectral gap and the corresponding asymptotic value are also presented. In [J. Graph Theory 99 (2022) 671-690], Horak and Kim established a sharp upper bound for the number of perfect matchings in terms of the Fibonacci number. We obtain a spectral characterization for the extremal graphs by showing that a cubic bipartite graph has the maximum number of perfect matchings if and only if it minimizes the algebraic connectivity.