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Graph Complements of Cyclic Graphs

Graph Complements of cyclic graphs have an astoundingly rich structure. They are homotopic to spheres or wedge sums of spheres. The graph complements of path graphs also are interesting as they are the unit spheres of graph complements of cyclic graphs,. They are always either spheres or contractible and carry a Gauss-Bonnet curvature which in the limit becomes a 6-periodic universal attractor. There is an other ``stable homotopy" manifestation visible when looking at the Lefschetz fixed point story. Graph complements of cyclic graphs have a dihedral symmetry group and so are Platonic in the sense that they define strongly regular geometries. The Lefschetz story for the symmetries shows a 12 periodicity in n. The Lefschetz numbers show patterns which repeat after n=12 steps which corresponds to a 4 periodicity in dimension. Besides the differential geometric, differential topological or algebraic topological point of view, the graphs are also exciting in graph theory per se, as one can essentially compute all graph quantities for them, Wiener index, Harary index, Tree-Forest ratios, chromatic number, eigenvalues of Laplacian or adjacency matrices, independence number, diameter, Shannon capacity. The combinatorics of cliques is related to Hyper-Fibonacci numbers (Lucas numbers), Jacobsthal polynomials, hyper Pascal triangles which make also the curvature expressions explicit. Here is a write-up on the Arxiv: https://arxiv.org/abs/2101.06873 and a blog entry https://www.quantumcalculus.org/graph... which also has links to the code to compute all these things: http://people.math.harvard.edu/~knill...