Abstract
Let $\text{Fix}(X,Y)$ be a semigroup of full transformations on a set $X$ in which elements in a nonempty subset $Y$ of $X$ are fixed. In this paper, we construct the Cayley digraphs of $\text{Fix}(X,Y)$ and study some structural properties of such digraphs such as the connectedness and the completeness. Further, some prominent results of Cayley digraphs of $\text{Fix}(X,Y)$ relative to minimal idempotents are verified. In addition, the characterization of an equivalence digraph of the Cayley digraph of $\text{Fix}(X,Y)$ is also investigated.
Highlights
In algebraic graph theory, the structures of algebraic methods are studied and applied to problems about graphs
A well-known connection between graphs and algebraic systems is the construction of Cayley graphs of groups
The Cayley graph was first introduced for finite groups by Arthur Cayley in 1878
Summary
The structures of algebraic methods are studied and applied to problems about graphs. We investigate the properties of the connectedness and the completeness of Cayley graphs Cay(Fix(X, Y ), A). Some structural properties of Cayley graphs of Fix(X, Y ) related to minimal idempotents are considered .
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