The new methods for constructing matching-equivalence graphs

Two graphs G and H with order n 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 and H for each r  , where 0⩽r⩽n0⩽r⩽n. In this paper, the new methods for constructing ‘matching-equivalent’ graphs are given, and some families of non-matching unique graphs are also obtained.