2023/01/17 by Bhowmik, Anwita, Barman, Rupam · 1 citation
#Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2301.07021
Let n=2s p1^α1⋯ pk^αk, where s=0 or 1, αi≥ 1, and the distinct primes pi satisfy pi≡ 1\pmod4 for all i=1, …, k. Let ℤn^∗ denote the group of units in the commutative ring ℤn. Recently, we defined a Paley-type graph Gn of order n as the graph whose vertex set is ℤn and xy is an edge if x-y≡ a2\pmod n for some a∈ℤn^∗. The Paley-type graph Gn resembles the classical Paley graph in a number of ways, and adds to the list of generalizations of the Paley graph. Computing the number of cliques of a particular order in a Paley graph or its generalizations has been of considerable interest. For primes p≡ 1\pmod 4 and α≥ 1, by evaluating certain character sums, we found the number of cliques of order 3 in Gpα and expressed the number of cliques of order 4 in Gpα in terms of Jacobi sums. In this article we give combinatorial proofs and find the number of cliques of orders 3 and 4 in Gn for all n for which the graph is defined.