Recursively constructible families of graphs

An infinite sequence of graphs {G"n}"n">="0 is called recursive if the Tutte polynomials T(G"n;x,y) satisfy a linear recurrence relation whose coefficients are polynomials in x and y. In this paper we introduce a general method based on transfer matrices for proving that a family is recursive that covers all examples known to us. As an application we show that, for fixed s, the sequence of complete bipartite graphs {K"s","n} is recursive and satisfies a linear recurrence whose degree is the number of partitions of the integer s.