Number of complete subgraphs of Peisert graphs and finite field hypergeometric functions

2022/05/08 by Bhowmik, Anwita, Barman, Rupam · 2 citations
#Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2205.03928

Abstract

For a prime p≡ 3\pmod4 and a positive integer t, let q=p^2t. Let g be a primitive element of the finite field \mathbbF_q. The Peisert graph P^∗(q) is defined as the graph with vertex set \mathbbF_q where ab is an edge if and only if a-b∈⟨ g⁴⟩ ∪ g⟨ g⁴⟩. We provide a formula, in terms of finite field hypergeometric functions, for the number of complete subgraphs of order four contained in P^∗(q). We also give a new proof for the number of complete subgraphs of order three contained in P^∗(q) by evaluating certain character sums. The computations for the number of complete subgraphs of order four are quite tedious, so we further give an asymptotic result for the number of complete subgraphs of any order m in Peisert graphs.

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