Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\displaystyle X_n$</EquationSource> <EquationSource Format="MATHML"><math> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mi>n</mi> </msub> </mstyle> </math></EquationSource> </InlineEquation> be a set of n elements and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\displaystyle T_n$</EquationSource> <EquationSource Format="MATHML"><math> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mi>n</mi> </msub> </mstyle> </math></EquationSource> </InlineEquation> the semigroup of full transformations on <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\displaystyle X_n$</EquationSource> <EquationSource Format="MATHML"><math> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mi>n</mi> </msub> </mstyle> </math></EquationSource> </InlineEquation> under the composition of functions. By <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\displaystyle \textrm{CR}(T_n)$</EquationSource> <EquationSource Format="MATHML"><math> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mtext>CR</mtext> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mstyle> </math></EquationSource> </InlineEquation> we denote the Cayley regularity graph of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\displaystyle T_n$</EquationSource> <EquationSource Format="MATHML"><math> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mi>n</mi> </msub> </mstyle> </math></EquationSource> </InlineEquation>, which we define as a digraph whose vertex set is <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\displaystyle T_n$</EquationSource> <EquationSource Format="MATHML"><math> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mi>n</mi> </msub> </mstyle> </math></EquationSource> </InlineEquation> and arc set contains all ordered pairs <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\displaystyle (\alpha , \beta )\in T_n\times T_n$</EquationSource> <EquationSource Format="MATHML"><math> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <msub> <mi>T</mi> <mi>n</mi> </msub> <mo>×</mo> <msub> <mi>T</mi> <mi>n</mi> </msub> </mrow> </mstyle> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\displaystyle \alpha = \alpha \beta \alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mi>α</mi> <mo>=</mo> <mi>α</mi> <mi>β</mi> <mi>α</mi> </mrow> </mstyle> </math></EquationSource> </InlineEquation>. We investigate structural properties of <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\displaystyle \textrm{CR}(T_n)$</EquationSource> <EquationSource Format="MATHML"><math> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mtext>CR</mtext> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mstyle> </math></EquationSource> </InlineEquation> by studying the questions of connectedness, completeness and traversability. We also consider the planarity of <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\displaystyle \textrm{CR}(T_n)$</EquationSource> <EquationSource Format="MATHML"><math> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mtext>CR</mtext> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mstyle> </math></EquationSource> </InlineEquation> by proving that <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\displaystyle \textrm{CR}(T_n)$</EquationSource> <EquationSource Format="MATHML"><math> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mtext>CR</mtext> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mstyle> </math></EquationSource> </InlineEquation> is planar if and only if n equals 2. Furthermore, we present inequalities for certain invariant parameters of <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\displaystyle \textrm{CR}(T_n)$</EquationSource> <EquationSource Format="MATHML"><math> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mtext>CR</mtext> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mstyle> </math></EquationSource> </InlineEquation>.

On structural and invariant properties of Cayley regularity graphs of full transformation semigroups

期刊论文
发表日期：

2025年6月26日

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Nuttawoot Nupo,
Chollawat Pookpienlert,
Yanisa Chaiya

摘要

Let $\displaystyle X_n$ $X_{n}$ be a set of n elements and $\displaystyle T_n$ $T_{n}$ the semigroup of full transformations on $\displaystyle X_n$ $X_{n}$ under the composition of functions. By $\displaystyle \textrm{CR}(T_n)$ $CR (T_{n})$ we denote the Cayley regularity graph of $\displaystyle T_n$ $T_{n}$ , which we define as a digraph whose vertex set is $\displaystyle T_n$ $T_{n}$ and arc set contains all ordered pairs $\displaystyle (\alpha , \beta )\in T_n\times T_n$ $(α, β) \in T_{n} \times T_{n}$ such that $\displaystyle \alpha = \alpha \beta \alpha $ $α = α β α$ . We investigate structural properties of $\displaystyle \textrm{CR}(T_n)$ $CR (T_{n})$ by studying the questions of connectedness, completeness and traversability. We also consider the planarity of $\displaystyle \textrm{CR}(T_n)$ $CR (T_{n})$ by proving that $\displaystyle \textrm{CR}(T_n)$ $CR (T_{n})$ is planar if and only if n equals 2. Furthermore, we present inequalities for certain invariant parameters of $\displaystyle \textrm{CR}(T_n)$ $CR (T_{n})$ .