Summary: Graph powers, Delsarte, Hoffman, Ramsey and Shannon
April 28, 2006
The k-th p-power of a graph G is the graph on the vertex set V (G)k
, where two k-tuples are
adjacent iff the number of their coordinates which are adjacent in G is not congruent to 0 modulo
p. The clique number of powers of G is poly-logarithmic in the number of vertices, thus graphs
with small independence numbers in their p-powers do not contain large homogenous subsets.
We provide algebraic upper bounds for the asymptotic behavior of independence numbers of
such powers, settling a conjecture of  up to a factor of 2. For precise bounds on some graphs,
we apply Delsarte's linear programming bound and Hoffman's eigenvalue bound. Finally, we
show that for any nontrivial graph G, one can point out specific induced subgraphs of large
p-powers of G with neither a large clique nor a large independent set. We prove that the larger
the Shannon capacity of G is, the larger these subgraphs are, and if G is the complete graph,
then some p-power of G matches the bounds of the Frankl-Wilson Ramsey construction, and is
in fact a subgraph of a variant of that construction.
The k-th Xor graph power of a graph G, Gk, is the graph whose vertex set is the cartesian product