Conclusions

In this paper we have extended the classes of graphs for which we know the mod sum number. These are

• $\rho(K_{n_1,n_2,...,n_m})>0$ if there exist $n_i$ and $n_j$ such that $n_i <n_j<2n_i$.
• $\rho(K_{n,m})= 0$ when $m \ge 2n$, $n \ge 3$ and m even.
• $\rho(K_{n,m})= 0$ when $m \ge 3n-3$, $n \ge 3$ and m odd.
• $\rho (K_{n,m})=n$ when $2n>m \ge n \ge 3$.

So we conclude with conjectures that our research has shown may be possible, but for lack of time we were not able to pursue.