On strongly 2-multiplicative graphs

A simple connected graph G of order n ≥ 3 is a strongly 2-multiplicative if there is an injective mapping f : V (G) ! f1; 2; : : : ; ng such that the induced mapping h : A ! Z+ dened by h(P) = Q3 i=1 f(vj i), where j1; j2; j3 2 f1; 2; : : : ; ng, and P is the path homotopy class of paths having the vertex set fvj1 ; vj2 ; vj3g, is injective. Let Δ(n) be the number of distinct path homotopy classes in a strongly 2-multiplicative graph of order n. In this paper we obtain an upper bound and also a lower bound for Δ(n). Also we prove that triangular ladder, P2 J Cn, Pm J Pn, the graph obtained by duplication of an arbitrary edge by a new vertex in path Pn and the graph obtained by duplicating all vertices by new edges in a path Pn are strongly 2-multiplicative.