$S$-normal Cayley graphs and hom-idempotent graphs
Authors :
Liny Mariam Mathew 1 *, G. N. Prakash 2 and L. John 3
Author Address :
1 Department of Mathematics, Devaswom Board College, Thalayolaparambu, Kottayam-686605, Kerala, India.
2 Department of Mathematics, Panampilly Memorial Government College, Chalakudy, Thrissur-680722, Kerala, India.
3 Department of Mathematics, University of Kerala, Thiruvananthapuram, Kerala, India.
*Corresponding autor.
Abstract :
Let $S$ be a semigroup and let $T$ be a subset of $S$. The Cayley graph $Cay(S, T)$ of $S$ relative to $T$ is defined as the graph with vertex set $S$ and edge set $E(T)$ consisting of those ordered pairs $(x, y)$ where $xt = y$ for some $t \in T$. $Cay(S, T)$ is said to be an $S$-normal Cayley graph if for all $z\in S$ and $t \in T$, $zt = t'z$ for some $t' \in T$. A graph $G$ is said to be hom-idempotent if there is a homomorphism from $G^2$ to $G$. In this paper we characterize hom-idempotent graphs in terms of $S$-normal Cayley graphs.
Keywords :
Semigroup, Cayley graph, $S$-normal Cayley graph, hom-idempotent graphs.
DOI :
Article Info :
Received : December 24, 2020; Accepted : February 27, 2021.
