Inequalities between distance-based graph polynomials

In a recent paper [ I. Gutman, Bull. Acad. Serbe Sci. Arts (Cl. Math. Natur) 131 (2005) 1-7], the Hosoya polynomial H = H(G,?) of a graph G, and two related distance-based polynomials H1 = H1(G, ?) and H2 = H2(G, ?) were examined. We now show that max-?H1 - ?2H, ?H1 - ?2H} ? H2 ? ?H1 - ??H holds for all graphs G and for all ? ? 0, where ? and ? are the smallest and greatest vertex degree in G. The answer to the question which of the terms ?H1 -?2H and ?H1 -?2H is greater, depends on the graph G and on the value of the variable ?. We find a number of particular solutions of this problem. AMS Mathematics Subject Classification (2000): 05C12, 05C05.