Scholarlink

<p>A perfect code <i>C</i> within a graph <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\( \Gamma \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation> consists of an independent set of vertices such that every vertex not in <i>C</i> is connected to exactly one vertex within <i>C</i>. A total perfect code <i>C</i> in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\( \Gamma \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation> is defined as a set of vertices where every vertex in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\Gamma \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation> is adjacent to a unique vertex in <i>C</i>. Let <i>G</i> represent a finite group, and let <i>X</i> be a normal subset of <i>G</i>. The Cayley sum graph <i>CS</i>(<i>G</i>,&#xa0;<i>X</i>) is constructed with vertex set <i>G</i>, where two vertices <i>g</i> and <i>h</i> are adjacent if <i>gh</i> is included in <i>X</i> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\( g \ne h \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mo>≠</mo> <mi>h</mi> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we demonstrate that a group <i>G</i> is abelian if and only if every subgroup of <i>G</i> qualifies as either a perfect code or a total perfect code in some Cayley sum graph associated with <i>G</i>.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

On perfect codes and total perfect codes in Cayley sum graph

  • 期刊论文

  • 出版模式：

    开放获取

  • 发表日期：

    2026年5月13日

查看全文
  • Neda Ahanjideh,
  • Zeinab Akhlaghi,
  • Bernardo G. Rodrigues

摘要

A perfect code C within a graph \( \Gamma \) Γ consists of an independent set of vertices such that every vertex not in C is connected to exactly one vertex within C. A total perfect code C in \( \Gamma \) Γ is defined as a set of vertices where every vertex in \(\Gamma \) Γ is adjacent to a unique vertex in C. Let G represent a finite group, and let X be a normal subset of G. The Cayley sum graph CS(G, X) is constructed with vertex set G, where two vertices g and h are adjacent if gh is included in X and \( g \ne h \) g ≠ h . In this paper, we demonstrate that a group G is abelian if and only if every subgroup of G qualifies as either a perfect code or a total perfect code in some Cayley sum graph associated with G.