正则图上生成与仿射生成 $\Gamma$-幻映射及 $\Gamma$-距离幻标定的代数刻画

中文摘要 AI 辅助

对于阶为 $n$ 的阿贝尔群 $\Gamma$，阶为 $n$ 的图 $G$ 称为 $\Gamma$-距离幻图，如果存在双射 $V(G)\to\Gamma$ 使得其开邻域和是常数；若对每个这样的 $\Gamma$ 该性质都成立，则称为群距离幻图。更一般地，对于任意有限阿贝尔群 $\Gamma$，$\Gamma$-幻映射是满足开邻域和为常数的映射 $f\colon V(G)\to\Gamma$；若其标号生成 $\Gamma$，则称 $f$ 为生成的；若其两两差生成 $\Gamma$，则称为仿射生成的。设 $G$ 为正则图，设 $\Gamma\cong\mathbb{Z}/d_1\oplus\cdots\oplus\mathbb{Z}/d_r$ 且 $d_1\mid\cdots\mid d_r$，令 $B_\Gamma=\bigoplus_{i<r}\mathbb{Z}/d_i$，并令 $\overline{A}_m$ 为 $A$ 在 $(\mathbb{Z}/m)^{V(G)}$ 上模常数诱导的邻接算子。我们证明：$G$ 存在生成的 $\Gamma$-幻映射当且仅当 $B_\Gamma\hookrightarrow\ker\overline{A}_{d_r}$，存在仿射生成的 $\Gamma$-幻映射当且仅当 $\Gamma\hookrightarrow\ker\overline{A}_{d_r}$；当 $|V(G)|=|\Gamma|$ 时，$G$ 是 $\Gamma$-距离幻图当且仅当后者嵌入的像具有顶点分离性。若约化邻接算子在 $\mathbb{Q}$ 上非奇异，则这些条件化为约化邻接 Smith 群中的子群条件。Cichacz 和 Froncek 猜想每个距离幻图都是群距离幻图。利用仿射判据，我们构造了一个 $27$ 阶的 $6$-正则距离幻图，它不存在仿射生成的 $(\mathbb{Z}/3)^3$-幻映射，因此该猜想即使在仿射生成意义下也失败。我们提出生成群距离幻猜想，并在 $\Gamma$ 为 $2$-生成（从而阶无立方因子）时对正则距离幻图证明之。进一步的应用涉及初等阿贝尔 $p$-群上的 Cayley 图、Hamming 关系图、强正则图以及对称设计。

英文摘要

For an abelian group $Γ$ of order $n$, a graph $G$ of order $n$ is $Γ$-distance magic if it admits a bijection $V(G)\toΓ$ whose open-neighborhood sums are constant, and group distance magic if this holds for every such $Γ$. More generally, for any finite abelian $Γ$, a $Γ$-magic map is a map $f\colon V(G)\toΓ$ with constant open-neighborhood sums; we call $f$ generating if its labels generate $Γ$, and affinely generating if its pairwise differences do. Let $G$ be regular, let $Γ\cong\mathbb{Z}/d_1\oplus\cdots\oplus\mathbb{Z}/d_r$ with $d_1\mid\cdots\mid d_r$, put $B_Γ=\bigoplus_{i<r}\mathbb{Z}/d_i$, and let $\overline{A}_m$ be the adjacency operator induced on $(\mathbb{Z}/m)^{V(G)}$ modulo constants. We prove that $G$ admits a generating $Γ$-magic map iff $B_Γ\hookrightarrow\ker\overline{A}_{d_r}$, and an affinely generating one iff $Γ\hookrightarrow\ker\overline{A}_{d_r}$; when $|V(G)|=|Γ|$, it is $Γ$-distance magic iff the latter embedding has vertex-separating image. If the reduced adjacency operator is nonsingular over $\mathbb{Q}$, these become subgroup conditions in the reduced adjacency Smith group. Cichacz and Froncek conjectured that every distance magic graph is group distance magic. Using the affine criterion we construct a $6$-regular distance magic graph of order $27$ admitting no affinely generating $(\mathbb{Z}/3)^3$-magic map, so the conjecture fails even for affine generation. We propose the generating group distance magic conjecture, and prove it for regular distance magic graphs whenever $Γ$ is $2$-generated, hence for cube-free order. Further applications concern Cayley graphs on elementary abelian $p$-groups, Hamming relation graphs, strongly regular graphs, and symmetric designs.

发表机构

• Izmir Institute of Technology（伊兹密尔理工大学）

机构由 AI 辅助整理，请以论文原文为准。