Matchings in bipartite graphs with a given number of cuts

Abstract

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 $m(G,k)\le m(H,k)$ holds for all $k$. Let ${\mathrm{BG}}_1(n,\gamma )$ be the set of connected bipartite graphs with $n$ vertices and $\gamma$ cut vertices, and ${\mathrm{BG}}_2(n,\gamma )$ 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 ${\mathrm{BG}}_1(n,\gamma )$ and the greatest element in ${\mathrm{BG}}_2(n,\gamma )$ 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.