Sum structures in abelian groups

Abstract

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′+Q′, 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 {Pi}in Zmclose up into a “Fibonacci cycle”–i.e. P0=1, P1=k, Pi+2=Pi+Pi+1for all integers i≥0, and then Pn=P0and Pn+1=P1–provided that m=Lnis a Lucas prime, in which case actually Pi=kifor all i≥0.

Keywords

• Graph with colored edges
• System of linear equations
• Sum cograph
• Catalogue
• Fibonacci cycle
• Lucas prime