On large arbitrary left path within a semigroup

Abstract

This paper is about the construction of semigroups $S$ from some given graph $G$. Let $S$ be a finite non-commutative semigroup, its commuting graph, denoted by $G(S)$, is a simple graph (which has no loops and multiple edges) whose sets of vertices are elements of $S$ and whose sets of edges are those elements of $S$ which commute with other elements i.e. for any $a,b \in S$ such that $ab=ba$ for $a \neq b$.For some non empty finite set $X$,denote $T(X)$ by semigroup of full transformations and $I_{r}$ by ideal of $T(X)$ whose rank is less than or equals to $r$. Let $a_{1}-a_{2}-a_{3}-\ldots-a_{m}$ be a path in $G(S)$, this path is said to be left path or $l-$ path if

$$a_{1}a_{i}=a_{m}a_{i} for i\in\{1,2,3,\ldots m\}$$

In this paper, we construct semigroup $S$ of a complete bipartite graph $K _{n,m}$ and find maximum length of $l-$ path in its commuting graph $G(S)$. Moreover,we see that such type of semigroups have knit degree $2$.