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Connected components and non-bipartiteness of generalized Paley graphs

Ricardo A. Podestá, Denis E. Videla

TL;DR

This work studies generalized Paley graphs $Γ(k,q)$, Cayley graphs on the finite field $F_q$ with connection set $igl\{x^k:x∈F_q^*\bigr\}$, addressing connected components and bipartiteness. The authors prove a subfield decomposition: if $Γ(k,p^m)$ is not connected, it is a disjoint union of $p^{m-a}$ copies of the smaller graph $Γ(k_a,p^a)$ where $a=ord_n(p)$, $n=(p^m-1)/k$, and $k_a=(p^a-1)/n$, with the components arising from the subfield $F_a$. They classify GP-graphs that are unions of cycles, showing that for odd $p$ the only such connected graphs are $Γ((q-1)/2,q)$ and $Γ(q-1,q)$, decomposing into $p^{m-1}$ copies of $C_p$ or $vec{C}_p$, respectively, and they provide exact automorphism groups and spectra. The paper also establishes a general non-bipartiteness result: $Γ(k,q)$ is non-bipartite except for the trivial case $Γ(2^m-1,2^m) ≅ 2^{m-1}K_2$, with implications for the connectedness of bipartite doubles and for various GP-graph families. Overall, the results give a unified structural and spectral framework for GP-graphs, enabling precise invariants and facilitating analysis across directed/undirected, cycle-decomposable, and semiprimitive cases.

Abstract

In this work we consider the class of Cayley graphs known as generalized Paley graphs (GP-graphs for short) given by $Γ(k,q) = Cay(\mathbb{F}_q, \{x^k : x\in \mathbb{F}_q^* \})$, where $\mathbb{F}_q$ is a finite field with $q$ elements, both in the directed and undirected case. Hence $q=p^m$ with $p$ prime, $m\in \mathbb{N}$ and one can assume that $k\mid q-1$. We first give the connected components of an arbitrary GP-graph. We show that these components are smaller GP-graphs all isomorphic to each other (generalizing a Lim and Praeger's result from 2009 to the directed case). We then characterize those GP-graphs which are disjoint unions of odd cycles. Finally, we show that $Γ(k,q)$ is non-bipartite except for the graphs $Γ(2^m-1,2^m)$, $m \in \mathbb{N}$, which are isomorphic to $K_2 \sqcup \cdots \sqcup K_2$, the disjoint union of $2^{m-1}$ copies of $K_2$.

Connected components and non-bipartiteness of generalized Paley graphs

TL;DR

This work studies generalized Paley graphs , Cayley graphs on the finite field with connection set , addressing connected components and bipartiteness. The authors prove a subfield decomposition: if is not connected, it is a disjoint union of copies of the smaller graph where , , and , with the components arising from the subfield . They classify GP-graphs that are unions of cycles, showing that for odd the only such connected graphs are and , decomposing into copies of or , respectively, and they provide exact automorphism groups and spectra. The paper also establishes a general non-bipartiteness result: is non-bipartite except for the trivial case , with implications for the connectedness of bipartite doubles and for various GP-graph families. Overall, the results give a unified structural and spectral framework for GP-graphs, enabling precise invariants and facilitating analysis across directed/undirected, cycle-decomposable, and semiprimitive cases.

Abstract

In this work we consider the class of Cayley graphs known as generalized Paley graphs (GP-graphs for short) given by , where is a finite field with elements, both in the directed and undirected case. Hence with prime, and one can assume that . We first give the connected components of an arbitrary GP-graph. We show that these components are smaller GP-graphs all isomorphic to each other (generalizing a Lim and Praeger's result from 2009 to the directed case). We then characterize those GP-graphs which are disjoint unions of odd cycles. Finally, we show that is non-bipartite except for the graphs , , which are isomorphic to , the disjoint union of copies of .
Paper Structure (4 sections, 6 theorems, 109 equations, 3 figures)

This paper contains 4 sections, 6 theorems, 109 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The connected components of the graphs $\Gamma(k,q)$
  3. GP-graphs as disjoint unions of cycles
  4. Non-bipartiteness of $\Gamma(k,q)$

Key Result

Theorem 2.1

Let $q=p^m$ be a prime power with $m\in \mathbb{N}$, $k \in \mathbb{N}$ such that $k\mid q-1$ and put $n=\frac{q-1}{k}$. Let $a=ord_n(p)$ and let $k_a = \frac{p^{a}-1}{n} \in\mathbb{N}$. Then, $a \mid m$, $k_a \mid k$, and $\Gamma=\Gamma(k,q)$ has exactly $p^{m-a}$ (strongly) connected components, a where $F_a = \{x \in \mathbb{F}_q : x^{p^a}=x\}$. Hence, the automorphism group of $\Gamma$ is give

Figures (3)

  • Figure 1: The graphs $\Gamma(1,9)$ and $\Gamma(2,9)$.
  • Figure 2: The graphs $\Gamma(4,9)$ and $\Gamma(8,9)$.
  • Figure 3: The Clebsch graph $\Gamma(3,16)$.

Theorems & Definitions (25)

  • Theorem 2.1
  • proof
  • Remark 2.2
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Example 2.6
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • ...and 15 more