Graph complements of circular graphs

TL;DR

This paper explores the properties of graph complements of cyclic graphs, revealing their topological, spectral, and combinatorial structures, including homotopy types, curvature, and periodicities, with implications for graph theory and topology.

Contribution

It provides a comprehensive analysis of the topological and spectral properties of graph complements of cyclic graphs, introducing new invariants and periodicity results.

Findings

Graph complements G(n) are circulant, vertex-transitive, claw-free, strongly regular, Hamiltonian, with Z(n) symmetry.

The forest-tree ratio in G(n) converges to e, and G(n) are Cayley graphs with Platonic properties.

Various topological invariants exhibit periodicity, such as 6-periodic Gauss-Bonnet curvature and 12-periodic Lefschetz numbers.

Abstract

Graph complements G(n) of cyclic graphs are circulant, vertex-transitive, claw-free, 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 forest-tree ratio converges to e. The graphs G(n) are Cayley graphs and so Platonic with isomorphic unit spheres G(n-3)^+, complements of path graphs. G(3d+3) are homotop to wedge sums of two d-spheres and G(3d+2),G(3d+4) are homotop to d-spheres, G(3d+1)^+ are contractible, G(3d+2)^+,G(3d+3)^+ are d-spheres. Since disjoint unions are dual to Zykov joins, graph complements of 1-dimensional 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 1-manifold 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 f-vectors 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 k-king configurations on a circular chess board. While the Euler curvature of circle complements G(n) is constant by symmetry, the discrete Gauss-Bonnet curvature of path complements G(n)^+ 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 G(n). There is also a 12-periodicity of Wu characteristic. This is a 4 periodicity in dimension.These are manifestations of stable homotopy features, but combinatorial.