Graph complements of circular graphs

In this report, we study graph complements Gn of cyclic graphs Cn or graph complements G + n of path graphs. Gn are circulant, vertex-transitive, claw-free, strongly regular, Hamiltonian graphs with a Zn symmetry and Shannon capacity 2. Also the Wiener and Harary index are known. The explicitly known adjacency matrix spectrum leads to explicit spectral zeta function and tree or forest quantities. The forest-tree ratio of Gn converge to e in the limit when n goes to infinity. The graphs Gn are all Cayley graphs and so Platonic in the sense that they all have isomorphic unit spheres G+n−3. The graphs G3d+3 are homotop to wedge sums of two d-spheres and G3d+2, G3d+4 are homotop to d-spheres, G + 3d+1 are contractible, G+3d+2, G + 3d+3 are homotop to d-spheres. Since disjoint unions are dual to Zykov joins, graph complements of all 1-dimensional discrete manifolds G are homotop to either a point, a sphere or a wedge sums of spheres. If the length of every connected component of a 1-manifold is not divisible by 3, the graph complement of G must be a sphere. In general, the graph complement of a forest is either contractible or a sphere. It also follows that all induced strict subgraphs of Gn are either contractible or homotop to spheres. The f-vectors Gn or G + n satisfy a hyper Pascal triangle relation, the total number of simplices are hyper Fibonacci numbers. The simplex generating functions are Jacobsthal polynomials, generating functions of k-king configurations on a circular chess board. While the Euler curvature of circle complements Gn is constant by symmetry, the discrete Gauss-Bonnet curvature of path complements Gn can be expressed explicitly from the generating functions. There is now a non-trivial 6-periodic Gauss-Bonnet curvature universality in the complement of Barycentric limits. The Brouwer-Lefschetz fixed point theorem produces a 12-periodicity of the Lefschetz numbers of all graph automorphisms of Gn. There is also a 12-periodicity of Wu characteristic. This corresponds to 4-periodicity in dimension as n → n+ 3 is homotop to a suspension. These are all manifestations of stable homotopy features, but purely combinatorial.

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