1. Sum structures in abelian groups
- Author
-
Robert Haas
- Subjects
Graph with colored edges ,System of linear equations ,Sum cograph ,Catalogue ,Fibonacci cycle ,Lucas prime ,Mathematics ,QA1-939 - 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.
- Published
- 2023
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