Sum structures in abelian groups

Any set $S$ of elements from an abelian group produces a graph with colored edges $G$(S), with its points the elements of $S$, and the edge between points $P$ and $Q$ assigned for its “color” the sum $P+Q$. Since any pair of identically colored edges is equivalent to an equation $P+Q=P^{\prime }+Q^{\prime }$, the geometric—combinatorial figure $G$(S) is thus equivalent to a system of linear equations. This article derives elementary properties of such “sum cographs”, including forced or forbidden configurations, and then catalogues the 54 possible sum cographs on up to 6 points. Larger sum cograph structures also exist: Points $\left\{P_i\right\}$ in $Z_m$ close up into a “Fibonacci cycle”–i.e. $P_0=1$, $P_1=k$, $P_{i+2}=P_i+P_{i+1}$ for all integers $i\ge 0$, and then $P_n=P_0$ and $P_{n+1}=P_1$–provided that $m=L_n$ is a Lucas prime, in which case actually $P_i=k^i$ for all $i\ge 0$.