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 S⊂N is the graph (S,E) with uv∈E if and only if u+v∈S. A graph G is said to be an mod sum graph if it is isomorphic to the sum graph of some S⊂Z M \{0} and all arithmetic performed modulo M where M≥|S|+1. The mod sum number ρ(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 ρ(H m,3)=m for m≥3, H m,n ∪ρK 1 is exclusive for m≥3 and n≥4 and $m(n-1) \leq \rho(H_{m,n})\leq \frac{1}{2} mn(n-1)$ for m≥3 and n≥4.