The distance d(u, v) between two vertices u and v of a graph G is equal to the length of a shortest path that connects u and v. Define WW (G, x) = 1/2 ∑ {a,b}⊆V (G) x d(a,b)+d(a,b), where d(G) is the greatest distance between any two vertices. In this paper the hyper-Wiener polynomials of the Cartesian product, composition, join and disjunction of graphs are computed.

A subset W \(\subseteq\) V (G) of a graph G is an independent semitotal dominating set G, abbreviated ISTd-set if and every element exactly distance 2 from at least one other W. The domination number denoted by \(\gamma\)it2(G), the minimum cardinality G. In this paper, we study concept in graphs investigate conditions for on which ISTd-sets exist. Further, join any two are examined. Consequent...

A join graph is the complete union of two arbitrary graphs. We give sufficient conditions for a join graph to be 1-factorizable. As a consequence of our results, the Hilton’s Overfull Subgraph Conjecture holds true for several subclasses of join graphs. © 2006 Elsevier B.V. All rights reserved.