We found a match
Your institution may have access to this item. Find your institution then sign in to continue.
- Title
On the Paley graph of a quadratic character.
- Authors
Mináč, Ján; Muller, Lyle; Nguyen, Tung T.; Tân, Nguyễn Duy
- Abstract
Paley graphs form a nice link between the distribution of quadratic residues and graph theory. These graphs possess remarkable properties which make them useful in several branches of mathematics. Classically, for each prime number p we can construct the corresponding Paley graph using quadratic and non-quadratic residues modulo p. Therefore, Paley graphs are naturally associated with the Legendre symbol at p which is a quadratic Dirichlet character of conductor p. In this article, we introduce the generalized Paley graphs. These are graphs that are associated with a general quadratic Dirichlet character. We will then provide some of their basic properties. In particular, we describe their spectrum explicitly. We then use those generalized Paley graphs to construct some new families of Ramanujan graphs. Finally, using special values of L-functions, we provide an effective upper bound for their Cheeger number. As a by-product of our approach, we settle a question raised in [Cramer et al.: The isoperimetric and Kazhdan constants associated to a Paley graph, Involve 9 (2016), 293–306] about the size of this upper bound.
- Subjects
CONGRUENCES &; residues; GRAPH theory; PRIME numbers; L-functions; GAUSSIAN sums; ISOPERIMETRIC inequalities
- Publication
Mathematica Slovaca, 2024, Vol 74, Issue 3, p527
- ISSN
0139-9918
- Publication type
Article
- DOI
10.1515/ms-2024-0040