Condensed Ricci Curvature on Paley Graphs and their Generalizations

TL;DR

This paper characterizes the connectivity of generalized Paley graphs, develops a vertex partitioning algorithm based on finite field structures, and derives explicit formulas for condensed Ricci curvature on these graphs.

Contribution

It provides a new characterization of connectivity in generalized Paley graphs and introduces a curvature formula leveraging finite field properties.

Findings

Connected components of generalized Paley graphs are precisely described.

A vertex partitioning algorithm exploits finite field vector space structure.

Explicit formulas for condensed Ricci curvature are derived for certain Paley graphs.

Abstract

We explore properties of generalized Paley graphs and we extend a result of Lim and Praeger by providing a more precise description of the connected components of disconnected generalized Paley graphs. This result leads to a new characterization of when generalized Paley graphs are disconnected. We also provide necessary and sufficient divisibility conditions for the multiplicative group of the prime subfield of certain finite fields to be contained in the multiplicative subgroup of nonzero $k$-th powers. This latter result plays a crucial role in our development of a sorting algorithm on generalized Paley graphs that exploits the vector space structure of finite fields to partition certain subsets of vertices in a manner that decomposes the induced bipartite subgraph between them into complete balanced bipartite subgraphs. As a consequence, we establish a matching condition between these subsets of vertices that results in an explicit formula for the condensed Ricci curvature on certain Paley graphs and their generalizations.