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)→{1,2,…,n} such that the induced mapping h:a→z+ defined by h(p)=∏3i=1f(vji), where j1,j2,j3∈{1,2,…,n}, and p is the path homotopy class of paths having the vertex set {vj1,vj2,vj3}, is injective. let λ(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 λ(n). also we prove that triangular ladder, p2⨀cn, pm⨀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.