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Orbits of finite-field hypergeometric functions and complete subgraphs of generalized Paley graphs

Dermot McCarthy and Mason Springfield

Vol. 17 (2024), No. 2, 355–362
Abstract

In a recent result of Dawsey and McCarthy, a formula for the number of complete subgraphs of order 4 of generalized Paley graphs is given in terms of a sum of finite-field hypergeometric functions. Via known transformation formulas for finite-field hypergeometric functions, many of the summands in their formula are equal. They construct a group action representing these transformations so that the number of summands that need to be evaluated is reduced to orbit representatives. We expand the group used by Dawsey and McCarthy, reducing by up to 80% the number of summands to be evaluated.

Keywords
hypergeometric functions, finite-field hypergeometric functions, Paley graphs
Mathematical Subject Classification
Primary: 11T24, 05C25
Secondary: 05C30, 33C99
Supplementary material

Code to list orbits of $X_k$

Milestones
Received: 26 September 2023
Revised: 20 February 2024
Accepted: 21 February 2024
Published: 20 May 2024

Communicated by Kenneth S. Berenhaut
Authors
Dermot McCarthy
Department of Mathematics & Statistics
Texas Tech University
Lubbock, TX
United States
Mason Springfield
Department of Mathematics & Statistics
Texas Tech University
Lubbock, TX
United States