Generalised Paley Graphs with a Product Structure

Abstract

A graph is Cartesian decomposable if it is isomorphic to a Cartesian product of (more than one) strictly smaller graphs, each of which has more than one vertex and admits no such decomposition. These smaller graphs are called the Cartesian-prime factors of the Cartesian decomposition, and were shown, by Sabidussi and Vizing independently, to be uniquely determined up to isomorphism. We characterise by their parameters those generalised Paley graphs which are Cartesian decomposable, and we prove that for such graphs, the Cartesian-prime factors are themselves smaller generalised Paley graphs. This generalises a result of Lim and the second author which deals with the case where all the Cartesianprime factors are complete graphs. These results contribute to the determination, by parameters, of generalised Paley graphs with automorphism groups larger than the 1-dimensional affine subgroups used to define them.Date: November 2016.

The weight distribution of irreducible cyclic codes associated with decomposable generalized Paley graphs

“…In Section 2, we consider the weight distribution of the code C = C(k, q) associated with the graph Γ = Γ(k, q) which is Cartesian decomposable. In this case it was proved by Pearce and Praeger ( [18]) that Γ = b Γ 0 for some fixed GP-graph Γ 0 . In Theorem 2.2, we show that the weight distribution of C can be computed from the corresponding one of the smaller code C 0 associated with Γ 0 .…”

“…, Γ t as in (5) with t > 1. Recently, Pearce and Praeger ( [18]) characterized those generalized Paley graphs which are Cartesian decomposable. They proved that a Cartesian decomposable GP-graph is a product of copies of a single graph, which is necessarily another GP-graph.…”

A reduction formula for Waring numbers through generalized Paley graphs

“…Let Γ be a simple GP-graph which is cartesian decomposable. By the characterization in [12], Γ is a product of copies of a single GP-graph. More precisely, if Γ is simple and connected the following conditions are equivalent:…”

“…Lim and Praeger studied their automorphism groups and characterized all GP-graphs which are Hamming graphs ( [8]). Also, Pearce and Praeger characterized all GP-graphs which are cartesian descomposable ( [12]). Both classic Paley graphs and GP-graphs have been used to find linear codes with good decoding properties ( [2], [7], [17]).…”

“…In Section 2, we consider GP-graphs Γ(k, q) which are cartesian decomposable. Using the characterization of these graphs in [12], in Theorem 2.2 we get a reduction formula for the associated Waring's numbers g(k, q) where q = p m with p prime and m = ab for certain integers a, b. In Section 3 we consider the case b prime.…”