Involutory Cayley graphs of polynomial and power series rings over the ring of integers modulo $n$

TL;DR

This paper studies the properties of involutory Cayley graphs constructed from polynomial and power series rings over the integers modulo n, revealing structural insights into these algebraic graph constructs.

Contribution

It introduces and analyzes involutory Cayley graphs for polynomial and power series rings over the integers modulo n, expanding understanding of their algebraic and combinatorial properties.

Findings

Characterization of adjacency conditions in these graphs

Structural properties of the graphs over different rings

Potential applications in algebraic graph theory

Abstract

Let $R$ be a commutative ring with identity. The involutory Cayley graph $\mathcal{G}(R)$ of $R$ is defined as the graph whose vertex set is the set of elements of $R$, where two vertices $a$ and $b$ are adjacent exactly when $(a-b)^2=1$. This paper investigates the properties of involutory Cayley graphs associated with polynomial and power series rings over the ring of integers modulo $n$.