In this note we give an explanation for two phenomena mentioned in the concluding remarks of “The matching polynomial of a polygraph” by Babić et al. The following results are obtained: 1. Although three matrices for given polygraphs defined in the above article in general have different orders, they determine the same recurrence relations for the matching polynomial of these polygraghs. 2. Under certain symmetry conditions, the order of the recurrence relations can be reduced by almost a half.