Graph complements of circular graphs
Abstract
Graph complements G(n) of cyclic graphs are circulant, vertextransitive, clawfree, strongly regular, Hamiltonian graphs with a Z(n) symmetry, Shannon capacity 2 and known Wiener and Harary index. There is an explicit spectral zeta function and tree or forest data. The foresttree ratio converges to e. The graphs G(n) are Cayley graphs and so Platonic with isomorphic unit spheres G(n3)^+, complements of path graphs. G(3d+3) are homotop to wedge sums of two dspheres and G(3d+2),G(3d+4) are homotop to dspheres, G(3d+1)^+ are contractible, G(3d+2)^+,G(3d+3)^+ are dspheres. Since disjoint unions are dual to Zykov joins, graph complements of 1dimensional discrete manifolds G are homotop to a point, a sphere or a wedge sums of spheres. If the length of every connected component of a 1manifold is not divisible by 3, the graph complement of G is a sphere. In general, the graph complement of a forest is either contractible or a sphere. All induced strict subgraphs of G(n) are either contractible or homotop to spheres. The fvectors G(n) 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 kking configurations on a circular chess board. While the Euler curvature of circle complements G(n) is constant by symmetry, the discrete GaussBonnet curvature of path complements G(n)^+ can be expressed explicitly from the generating functions. There is now a nontrivial 6 periodic GaussBonnet curvature universality in the complement of Barycentric limits. The BrouwerLefschetz fixed point theorem produces a 12periodicity of the Lefschetz numbers of all graph automorphisms of G(n). There is also a 12periodicity of Wu characteristic. This is a 4 periodicity in dimension.These are manifestations of stable homotopy features, but combinatorial.
 Publication:

arXiv eprints
 Pub Date:
 January 2021
 DOI:
 10.48550/arXiv.2101.06873
 arXiv:
 arXiv:2101.06873
 Bibcode:
 2021arXiv210106873K
 Keywords:

 Mathematics  Combinatorics;
 Computer Science  Discrete Mathematics;
 57M15;
 68R10;
 05C50
 EPrint:
 47 pages, 28 figures