Related papers: Paley Graphs and Their Generalizations
Let $q$ be a prime power such that $q\equiv 1\pmod{4}$. The Paley graph of order $q$ is the graph with vertex set as the finite field $\mathbb{F}_q$ and edges defined as, $ab$ is an edge if and only if $a-b$ is a non-zero square in…
The Paley graph is a well-known self-complementary pseudo-random graph, defined over a finite field of odd order. We describe an attempt at an analogous construction using fields of even order. Some properties of the graph are noted, such…
In this work we consider the class of Cayley graphs known as generalized Paley graphs (GP-graphs for short) given by $\Gamma(k,q) = Cay(\mathbb{F}_q, \{x^k : x\in \mathbb{F}_q^* \})$, where $\mathbb{F}_q$ is a finite field with $q$…
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…
We consider a special class of generalized Paley graphs over finite fields, namely the Cayley graphs with vertex set $\mathbb{F}_{q^m}$ and connection set the nonzero $(q^\ell+1)$-th powers in $\mathbb{F}_{q^m}$, as well as their…
Let $k \geq 2$ be an integer. Let $q$ be a prime power such that $q \equiv 1 \pmod {k}$ if $q$ is even, or, $q \equiv 1 \pmod {2k}$ if $q$ is odd. The generalized Paley graph of order $q$, $G_k(q)$, is the graph with vertex set…
There is a Paley graph for each prime power $q$ such that $q\equiv 1\pmod 4$. The vertex set is the field $\mathbb Fq$ and two vertices $x$ and $y$ are joined by an edge if and only if $x-y$ is a nonzero square of $\mathbb Fq$. We compute…
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…
Given a finite field, one can form a directed graph using the field elements as vertices and connecting two vertices if their difference lies in a fixed subgroup of the multiplicative group. If -1 is contained in this fixed subgroup, then…
The generalised Paley graphs are, as their name suggests, a generalisation of the Paley graphs, first defined by Paley in 1933 (see \cite{Paley}). They arise as the relation graphs of symmetric cyclotomic association schemes. However, their…
A generalised Paley map is a Cayley map for the additive group of a finite field F, with a subgroup S=-S of the multiplicative group as generating set, cyclically ordered by powers of a generator of S. We characterise these as the…
As a vital link between group theory and graph theory, Cayley graphs provide a geometric framework for encoding algebraic structures. This study explores the properties of Cayley graphs derived from cyclic groups whose order is the square…
We investigate structural and combinatorial properties of Bi-Cayley graphs defined over cyclic groups of order $p^2q^2$, where $p$ and $q$ are distinct primes. We begin by describing their fundamental group-theoretic underpinnings. The main…
Paley graphs and Paley sum graphs are classical examples of quasi-random graphs. In this paper, we provide new constructions of families of quasi-random graphs that behave like Paley graphs but are neither Cayley graphs nor Cayley sum…
The family of generalized Paley graphs of prime power order $q$ and degree $(q-1)/k$ is studied. It is shown that the automorphism group of a graph in this family is a subgroup of ${\mathrm{A\Gamma L}}(1,q)$ whenever $q$ is sufficiently…
A simple undirected graph is said to be {\em semisymmetric} if it is regular and edge-transitive but not vertex-transitive. Every semisymmetric graph is a bipartite graph with two parts of equal size. It was proved in [{\em J. Combin.…
A graph $\G$ is {\em symmetric} or {\em arc-transitive} if its automorphism group $\Aut(\G)$ is transitive on the arc set of the graph, and $\G$ is {\em basic} if $\Aut(\G)$ has no non-trivial normal subgroup $N$ such that the quotient…
We consider the family of generalized Paley graphs (GP-graphs for short) $\Gamma(k,q) = Cay(\mathbb{F}_q, (\mathbb{F}_q^*)^k)$, with $q=p^m$ and $p$ prime. We characterize all GP-graphs having real spectrum; namely, $Spec(\Gamma(k,q))…
A graph is Cartesian decomposable if it is isomorphic to a Cartesian product of (more than one) strictly smaller graphs, each of which has more than one vertex and admits no such decomposition. These smaller graphs are called the…
We study the spectrum of generalized Paley graphs $\Gamma(k,q)=Cay(\mathbb{F}_q,R_k)$, undirected or not, with $R_k=\{x^k:x\in \mathbb{F}_q^*\}$ where $q=p^m$ with $p$ prime and $k\mid q-1$. We first show that the eigenvalues of…