Hosoya Polynomials of Steiner Distance of Some Graphs

The Steiner n-distance, d(S), of a non-empty n- subset S of vertices of a graph G is defined to be the size of the smallest connected subgraph T(S) containing S. The Hosoya polynomial of Steiner n- distance of a connected graph G is denoted by Hn* (G;x). In this work, we obtain Hosoya polynomials of Steiner n-distance(n is greater than or equal to 3 and less than or equal to the order of the graph) of some particular graphs; for other prescribed graphs, we obtain Hosoya polynomials of Steiner 3- distance. For some graphs G, we find reduction formulas for Hn*(G;x) or H3*(G;x). Wiener indices of the Steiner n-distance of most of the particular graphs and composite graphs considered here are also obtained. Moreover, the diameter of the Steiner n-distance for each one of these graphs is determined. Furthermore, Wiener index theorem for trees, which is due to H. Wiener, is generalized to Steiner n- distance of trees.