On {1, 2}-Factor Polynomial of Graphs

Many polynomials have been associated to graphs, such as the characteristic, matchings, chromatic and Tutte polynomials. Although these polynomials are inherent interesting, they encode useful combinatorial information about the given graph. So, it is natural then to ask to what extent any of these polynomials determines a graph and, in particular, whether one can find graphs that can be uniquely determined by a given polynomial. Let G be a graph of order n. We define the {1, 2}-factor polynomial of the graph G as f(G, x) = Σ0≤i≤n aixn-i, where ai, i = 1, …, n, is the number of {1, 2}-subgarphs of G of order i, and a0=1. In this paper, we gather some results on this polynomial. Specially, we prove that if G and H are two 4-cycle free graphs such that G is an r-regular graph and f(G, x) = f(H, x), then H is an r-regular graph.