Matchings in bipartite graphs with a given number of cuts

A matching in a graph is a set of pairwise nonadjacent edges. Denote by m(G, k) the number of matchings of cardinality k in a graph G. A quasi-order <= is defined by G <= H whenever m(G, k) < m(H, k) holds for all k. Let BG1(n, gamma ) be the set of connected bipartite graphs with n vertices and gamma cut vertices, and BG(2)(n, gamma ) be the set of connected bipartite graphs with n vertices and gamma cut edges. We determine the greatest and least elements with respect to this quasi-order in BG1(n, gamma ) and the greatest element in BG(2)(n, gamma ) for all values of n and gamma . As corollaries, we find that these graphs maximize (resp. minimize) the Hosoya index and the matching energy within the respective sets. (c) 2024 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.