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Periodic orbits on 2-regular circulant digraphs

Isaac Echols, Jon Harrison, Tori Hudgins

TL;DR

The paper addresses counting primitive periodic orbits on connected $2$-regular circulant digraphs, a problem tied to Ihara zeta functions and quantum-graph trace formulas. It develops a lattice-based approach to orbit lengths and step counts via the diophantine equation $la+kd=oldsymbol{\omega} n$ and leverages Lyndon words to count primitive step sequences, yielding explicit Möbius-inversion formulas for orbit counts. The main contributions are exact expressions for the number of primitive periodic orbits of fixed length and fixed $b$-count, and their total counts, expressed through binomial coefficients and divisors constrained by gcd and winding-number considerations. These results, complemented by a detailed example, provide a robust combinatorial framework that can feed into exact spectral-statistics calculations for quantum circulant graphs.

Abstract

Periodic orbits (equivalence classes of closed paths up to cyclic shifts) play an important role in applications of graph theory. For example, they appear in the definition of the Ihara zeta function and exact trace formulae for the spectra of quantum graphs. Circulant graphs are Cayley graphs of $\mathbb{Z}_n$. Here we consider directed Cayley graphs with two generators (2-regular Cayley digraphs). We determine the number of primitive periodic orbits of a given length (total number of directed edges) in terms of the number of times edges corresponding to each generator appear in the periodic orbit (the step count). Primitive periodic orbits are those periodic orbits that cannot be written as a repetition of a shorter orbit. We describe the lattice structure of lengths and step counts for which periodic orbits exist and characterize the repetition number of a periodic orbit by its winding number (the sum of the step sequence divided by the number of vertices) and the repetition number of its step sequence. To obtain these results, we also evaluate the number of Lyndon words on an alphabet of two letters with a given length and letter count.

Periodic orbits on 2-regular circulant digraphs

TL;DR

The paper addresses counting primitive periodic orbits on connected -regular circulant digraphs, a problem tied to Ihara zeta functions and quantum-graph trace formulas. It develops a lattice-based approach to orbit lengths and step counts via the diophantine equation and leverages Lyndon words to count primitive step sequences, yielding explicit Möbius-inversion formulas for orbit counts. The main contributions are exact expressions for the number of primitive periodic orbits of fixed length and fixed -count, and their total counts, expressed through binomial coefficients and divisors constrained by gcd and winding-number considerations. These results, complemented by a detailed example, provide a robust combinatorial framework that can feed into exact spectral-statistics calculations for quantum circulant graphs.

Abstract

Periodic orbits (equivalence classes of closed paths up to cyclic shifts) play an important role in applications of graph theory. For example, they appear in the definition of the Ihara zeta function and exact trace formulae for the spectra of quantum graphs. Circulant graphs are Cayley graphs of . Here we consider directed Cayley graphs with two generators (2-regular Cayley digraphs). We determine the number of primitive periodic orbits of a given length (total number of directed edges) in terms of the number of times edges corresponding to each generator appear in the periodic orbit (the step count). Primitive periodic orbits are those periodic orbits that cannot be written as a repetition of a shorter orbit. We describe the lattice structure of lengths and step counts for which periodic orbits exist and characterize the repetition number of a periodic orbit by its winding number (the sum of the step sequence divided by the number of vertices) and the repetition number of its step sequence. To obtain these results, we also evaluate the number of Lyndon words on an alphabet of two letters with a given length and letter count.

Paper Structure

This paper contains 6 sections, 17 theorems, 27 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Circulant Graphs
  3. The Lattice of Periodic Orbits
  4. Lyndon Words of Fixed Letter Count
  5. Counting Periodic Orbits
  6. Concluding Remarks

Key Result

theorem 1

Let $C_n(a,b)$ be a connected $2$-regular circulant digraph, and suppose $la + k(b-a) = \omega n$. The number of primitive periodic orbits of length $l$ with $b$-count $k$ is, where $\mu$ is the Möbius function.

Figures (6)

  • Figure 1: Examples of directed circulant graphs
  • Figure 2: The lattice $\bar{S}_7(1,3)$.
  • Figure 3: The periodic orbits $\varphi(w,0)$ for each $w\in\mathbb{L}_2(9,3)$, where $G = C_9(1,4)$. Repeated bonds are shown twice. The bonds are the same in $\varphi(111111444,0)$ and $\varphi(111141414,8)$, but these orbits are distinct due to the orders of step sequences.
  • Figure 4: The periodic orbit $\varphi(114114114,0)$ on $C_9(1,4)$, which is a primitive periodic orbit with a nonprimitive step-sequence.
  • Figure 5: Divisors $t = qm$ of $\gcd(360,240)=120$ with $q$ coprime with the repetition number $\omega=9$ and $m$ a square-free divisor of $\gcd(360/q,240/q)$. Arrows show two divisors are related by multiplication by a prime.
  • ...and 1 more figures

Theorems & Definitions (28)

  • theorem 1
  • theorem 2
  • theorem 3
  • proposition 1
  • proof
  • proposition 2
  • proof
  • corollary 1
  • lemma 1
  • proof
  • ...and 18 more