On the mod sum number of H m,n

Let N denote the set of all positive integers. The sum graph G (+)(S) of a finite subset SaS,N is the graph (S,E) with uvaE if and only if u+vaS. A graph G is said to be an mod sum graph if it is isomorphic to the sum graph of some SaS,Z (M) \{0} and all arithmetic performed modulo M where M >=|S|+1. The mod sum number rho(G) of G is the smallest number of isolated vertices which when added to G result in a mod sum graph. It is known that the graphs H (m,n) (n > m >= 3) are not mod sum graphs. In this paper we show that H (m,n) are not mod sum graphs for m >= 3 and n >= 3. Additionally, we prove that rho(H (m,3))=m for m >= 3, H (m,n) a(a)rho K (1) is exclusive for m >= 3 and n >= 4 and m(n - 1) <= rho(H (m,n)) <= 1/2 mn (n - 1) for ma >= 3 and n >= 4.