Level Three Arithmetic Graph Sums
Sam Vandervelde · 2008
A level three graph sum S(G) is the sum over all three-colorings of the vertices of a given multigraph G of the “total edge value ” for each coloring. Each edge is assigned a cube root of unity depending upon the colors of its endpoints; the product over all edges gives the corresponding term in the sum. This assignment may be defined via a symmetric 3 × 3 matrix whose entries are cube roots of unity. We demonstrate that the 729 possible matrices lead to 31 nonequivalent level three graph sums, then pinpoint 81 of them (representing 7 equivalence classes) with the nice property that S(G) gives a power of three for every graph G; these are the arithmetic graph sums. We propose three characterizations of matrices that give arithmetic graph sums, then prove that these seven graph sums have the desired property.