An infinite family of graphs with a facile count of perfect matchings

Given a graph G=(V,E), let the triangulationG△=(V△,E△)ofG be the graph obtained from G by supplementing each uv∈E with a new vertex w along with new edges uw and wv (while retaining uv). Let dv be the degree of a vertex v∈V and let G be a tree T. Then it is proved that the count of perfect matchings of the Cartesian product of T△ with K2 is given as the product of factors dv+1 over all v∈V. Also under favorable conditions, the degree sequence of T△×K2 is reconstructed via factorization of the number of its perfect matchings. Previously introduced degree product polynomials play a helpful role.