Cubic bipartite graphs with minimum spectral gap

TL;DR

This paper investigates the minimal algebraic connectivity and spectral gap of cubic bipartite graphs, providing asymptotic values, characterizations of extremal graphs, and linking these properties to the maximum number of perfect matchings.

Contribution

It establishes the asymptotic minimum algebraic connectivity for cubic bipartite graphs and characterizes the unique extremal graph, connecting spectral properties with combinatorial matchings.

Findings

Minimum algebraic connectivity is asymptotically (1+o(1))π²/n².

The unique cubic bipartite graph with minimum algebraic connectivity is characterized.

Maximum number of perfect matchings corresponds to graphs with minimum algebraic connectivity.

Abstract

The difference between the two largest eigenvalues of the adjacency matrix of a graph $G$ is called the spectral gap of $G.$ If $G$ is a regular graph, then its spectral gap is equal to algebraic connectivity. Abdi, Ghorbani and Imrich, in [European J. Combin. 95 (2021) 103328], showed that the minimum algebraic connectivity of cubic connected graphs on $2n$ vertices is $(1+o(1))\frac{\pi^{2}}{2n^{2}}$, which is attained on non-bipartite graphs. Motivated by the above result, we in this paper investigate the algebraic connectivity of cubic bipartite graphs. We prove that the minimum algebraic connectivity of cubic bipartite graphs on $2n$ vertices is $(1+o(1))\frac{\pi^{2}}{n^{2}}$. Moreover, the unique cubic bipartite graph with minimum algebraic connectivity is completed characterized. 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.