A bijection between aperiodic palindromes and connected circulant graphs

TL;DR

This paper establishes a one-to-one correspondence between palindromes and circulant graphs, and between aperiodic palindromes and connected circulant graphs, enabling enumeration of these graph classes based on palindrome properties.

Contribution

It introduces a novel bijection linking palindromic structures to circulant graphs, providing a new combinatorial perspective and enumeration method.

Findings

Bijection between palindromes and circulant graphs

Enumeration formulas for connected circulant graphs

Connection between aperiodic palindromes and connected graphs

Abstract

In this paper, we show that there is a one-to-one correspondence between the set of compositions (resp. prime compositions) of $n$ and the set of circulant digraphs (resp. connected circulant digraphs) of order $n$. We also show that there is a one-to-one correspondence between the set of palindromes (resp. aperiodic palindromes) of $n$ and the set of circulant graphs (resp. connected circulant graphs) of order $n$. As a corollary of this correspondence, we enumerate the number of connected circulant (di)graphs of order $n$.