Mod Sum Graph Definitions and Survey

Boland, Laskar, Turner and Domke introduced in [#!bltd!#] a generalization of sum graph. A graph is a mod sum graph if there exist a positive integer $\cal Z$ and a labelling of the vertices with distinct elements of $\{1,2,...,{\cal{Z}}-1\}$ such that $\{u,v\} \in E$ if and only if $u \ne v$ and $u+v$ $(mod$ ${\cal {Z}}) \in \cal F$. We similarly define the mod sum number of a graph $\rho(G)$ as the smallest number of isolated vertices that must be added to $G$ such that the resulting graph is a mod sum graph (MSG). So far the classifications on graphs are

• $\rho (K_n) =n$ for $n \ge 4$ [#!ms!#]
• $\rho (K_{2,n})=0$ and $\rho (K_{1,n})=0$ for $n \ge 2$ [#!bltd!#]
• $\rho (K_{p+1,q})=0$ for $q \ge 1$ and $p \ge r_q+ r_{q-1}-1$ where $r_1=1$, $r_2=2$, and $r_j=r_{j-2}+r_{j-1}+1$ for $j \ge3$ [#!bltd!#]
• $\rho (K_{n,n})>0$ for $n \ge 3$ [#!mrss!#]
• $\rho (H_{2,n})=0$ for $n \ge 2$ [#!bltd!#]
• $\rho (H_{m,n}) >0$ for $m >n \ge 2$ [#!ms!#]
• $\rho (C_n)=0$ for $n \ne 4$ [#!bltd!#]
• $\rho (W_n) > 0$ for $n\ge5$ [#!mrss!#]
• $\rho (T_n)=0$ for all $n \ge 3$ [#!bltd!#]