This paper presents a detailed study of the order structured conjugacy class simple graph (OSCSG) on the dihedral group, focusing on the key-theoretic parameters such as degree, average distance, rank, clique number, and independence number. We establish the criteria for the completeness and regularity of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Gamma ^{o}_{cl}(D_{2n})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi mathvariant="normal">Γ</mi> <mrow> <mi mathvariant="italic">cl</mi> </mrow> <mi>o</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>D</mi> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. For example, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\Gamma ^{o}_{cl}(S_{3})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi mathvariant="normal">Γ</mi> <mrow> <mi mathvariant="italic">cl</mi> </mrow> <mi>o</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>S</mi> <mn>3</mn> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is isomorphic to <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$K_{3}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>K</mi> <mn>3</mn> </msub> </math></EquationSource> </InlineEquation>. Further, we demonstrate that the domination number of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\Gamma ^{o}_{cl}(D_{2n})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi mathvariant="normal">Γ</mi> <mrow> <mi mathvariant="italic">cl</mi> </mrow> <mi>o</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>D</mi> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is 1 if n is even and 2 otherwise. The rank of OSCSG for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$G\cong D_{2n}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mo>≅</mo> <msub> <mi>D</mi> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </msub> </mrow> </math></EquationSource> </InlineEquation> is consistently 3. We observe that the complement graph of the OSCSG defined on <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$D_{2n}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>D</mi> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> is isomorphic to <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$2K_{2} \oplus K_{\frac{n-2}{2}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <msub> <mi>K</mi> <mn>2</mn> </msub> <mo>⊕</mo> <msub> <mi>K</mi> <mfrac> <mrow> <mi>n</mi> <mo>-</mo> <mn>2</mn> </mrow> <mn>2</mn> </mfrac> </msub> </mrow> </math></EquationSource> </InlineEquation> for even values of n. Our findings contribute to the optimization of cryptographic designs that enhance the efficiency of secure communication.

A new approach to conjugacy class simple graphs for the dihedral groups

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发表日期：

2025年9月9日

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Aneela,
Muhammad Khalid Mahmood,
Daud Ahmad

摘要

This paper presents a detailed study of the order structured conjugacy class simple graph (OSCSG) on the dihedral group, focusing on the key-theoretic parameters such as degree, average distance, rank, clique number, and independence number. We establish the criteria for the completeness and regularity of $\Gamma ^{o}_{cl}(D_{2n})$ $Γ_{cl}^{o} (D_{2 n})$ . For example, $\Gamma ^{o}_{cl}(S_{3})$ $Γ_{cl}^{o} (S_{3})$ is isomorphic to $K_{3}$ $K_{3}$ . Further, we demonstrate that the domination number of $\Gamma ^{o}_{cl}(D_{2n})$ $Γ_{cl}^{o} (D_{2 n})$ is 1 if n is even and 2 otherwise. The rank of OSCSG for $G\cong D_{2n}$ $G ≅ D_{2 n}$ is consistently 3. We observe that the complement graph of the OSCSG defined on $D_{2n}$ $D_{2 n}$ is isomorphic to $2K_{2} \oplus K_{\frac{n-2}{2}}$ $2 K_{2} \oplus K_{\frac{n - 2}{2}}$ for even values of n. Our findings contribute to the optimization of cryptographic designs that enhance the efficiency of secure communication.