Steiner Gutman Index

The concept of Gutman index SGut(G) of a connected graph G was introduced in 1994. The Steiner distance in a graph, introduced by Chartrand et al. in 1989, is a natural generalization of the concept of classical graph distance. In this paper, we generalize the concept of Gutman index by Steiner distance. The Steiner Gutman k-index SGut(k)(G) of G is defined by SGut(k)(G) = Sigma(s subset of v(G)vertical bar S vertical bar = k) [Pi(v is an element of S) deg(G)(v)] d(G)(S) where d(G)(S) is the Steiner distance of S and deg(G)(v) is the degree of v in G. Expressions for SGut(k) for some special graphs arc obtained. We also give sharp upper and lower bounds of SGut(k) of a connected graph, and get the expression of SGut(k)(G) for k = n, n - 1. Finally, we compare between k-center Steiner degree distance SDDk and SGut(k) of graphs.