  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.
