A class of ‘matching-equivalent’ bipartite graphs

Abstract

Two bipartite graphs G 1=(V 1=S 1 ∪ ̇ T 1,E 1) and G 2=(V 2=S 2 ∪ ̇ T 2,E 2) in which there are no isolated points and in which the cardinalities of the ‘upper’ sets are equal, that is, | S 1|=| S 2|= n (say), are said to be matching-equivalent if and only if the number of r-matchings (i.e., the number of ways in which r disjoint edges can be chosen) is the same for each of the graphs G 1 and G 2 for each r, 1⩽r⩽n . We show that the number of bipartite graphs that are matching-equivalent to K n, n , the complete bipartite graph of order ( n, n) is 2 n−1 subject to an inclusion condition on the sets of neighbors vertices of the ‘upper set’. The proof involves adding an arbitrary number of vertices to the ‘lower’ set which are neighbors to all the vertices in the upper set and then analyzing the ‘modified’ rook polynomial that is specially defined for the purpose of the proof.