On the theory of matching equivalent graphs and rook equivalent chessboards

Abstract

An investigation is made into graphs which have the same number of k-matchings, for all values of k (i.e. matching equivalent graphs). Some general constructions are presented. The results are then applied to the construction of chessboards which have the same rook polynomial (i.e. rook equivalent chessboards). This novel approach to problems on equivalence of chessboards (using the matching polynomial) seems to have some potential in the study of rook polynomials and, in general, permutations with restricted positions.