The diagonal graph

TL;DR

This paper introduces the diagonal graph, a vertex-transitive Cayley graph related to diagonal semilattices, and explores its properties, spectrum, and potential significance in algebraic graph theory.

Contribution

It establishes fundamental properties of the diagonal graph, including clique and chromatic numbers, spectrum, and symmetry, expanding understanding of diagonal groups beyond classical theorems.

Findings

Diagonal graph is vertex-transitive and a Cayley graph of G^m.

The clique number and chromatic number are determined in general and most cases.

Spectrum of the adjacency matrix is computed using the M"obius function.

Abstract

According to the O'Nan--Scott Theorem, a finite primitive permutation group either preserves a structure of one of three types (affine space, Cartesian lattice, or diagonal semilattice), or is almost simple. However, diagonal groups are a much larger class than those occurring in this theorem. For any positive integer $m$ and group $G$ (finite or infinite), there is a diagonal semilattice, a sub-semilattice of the lattice of partitions of a set $\Omega$, whose automorphism group is the corresponding diagonal group. Moreover, there is a graph (the diagonal graph), bearing much the same relation to the diagonal semilattice and group as the Hamming graph does to the Cartesian lattice and the wreath product of symmetric groups. Our purpose here, after a brief introduction to this semilattice and graph, is to establish some properties of this graph. The diagonal graph $\Gamma_D(G,m)$ is a Cayley graph for the group~$G^m$, and so is vertex-transitive. We establish its clique number in general and its chromatic number in most cases, with a conjecture about the chromatic number in the remaining cases. We compute the spectrum of the adjacency matrix of the graph, using a calculation of the M\"obius function of the diagonal semilattice. We also compute some other graph parameters and symmetry properties of the graph. We believe that this family of graphs will play a significant role in algebraic graph theory.