发表机构
University of Ngaoundéré; Laboratory of Energy, Signal, Imaging and Automation (LESIA), University of Ngaoundéré; Laboratory of Scientific Artificial Intelligence and Applied Mathematics, University of Garoua（恩加乌代雷大学; 恩加乌代雷大学能源、信号、成像与自动化实验室; 加鲁阿大学科学人工智能与应用数学实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究图成为阿贝尔群凯莱图的差距，定义γ⁺和γ△两个不变量，证明编辑版本问题NP完全，确定路径、网格等的不变量值，普查小顶点连通图并分析不变量特性。
AI 中文摘要
有限连通图很少是凯莱图，本文衡量其成为凯莱图的差距：给定含n个顶点、m条边的图G，求最少需添加边，或添加与删除边，使结果成为同一顶点集上n阶阿贝尔群的凯莱图，由此定义两个不变量：完备数γ⁺（仅添加边）和凯莱编辑距离γ△（添加与删除边），均以m归一化。本文证明，即使针对固定循环宿主，判定编辑版本问题也是NP完全的，方法是从哈密顿圈问题归约，当标签实现含k条边的最长路径时，其编辑代价为n+m−2k，最优代价为m−n+2pp(G)，可通过匹配数多项式时间界定。本文证明仅不规则性就迫使γ⁺(G)≥nΔ*/(2m)−1，其中Δ*是满足nd为偶数的最小d≥Δ，可从度序列线性时间计算，并刻画等号成立的精确条件；该等号在星图上实现，γ⁺(K₁,q)=(q−1)/2，且星图使γ⁺最大化，而γ△则被绝对常数界定。本文精确确定路径和网格的γ⁺，即γ⁺(Pₙ)=γ⁺(Pₙ□Pₙ)=1/(n−1)，并证明γ△(K₁,q)趋近于2，而非加法情况暗示的3/2。本文报告对所有至多7个顶点的995个连通图的详尽认证普查，度界在89.4%的图上实现，两个不变量在84.7%的图上严格分离，尽管这两个比率随阶数剧烈变化：n=4、5、6、7时，实现率分别为100%、100%、84.8%、89.7%，分离率分别为0%、61.9%、73.2%、87.7%，其中7个顶点的853个图占主导；星图唯一使两个不变量都最大化。编辑数与完备宿主的双利普希茨失真相互独立，在星图上反向变化，代码和认证见doi: https://doi.org/10.5281/zenodo.21852006。
英文摘要
A finite connected graph is rarely a Cayley graph. We measure how far it is from being one: given $G$ with $n$ vertices and $m$ edges, how few edges must be added, or added and deleted, before the result is a Cayley graph of an abelian group of order $n$ on the same vertex set? This defines two invariants, the completion number $γ^{+}$ (additions only) and the Cayley edit distance $γ_{\triangle}$ (both), each normalized by $m$. We show that deciding the edit version is NP-complete already for a fixed cyclic host, by a reduction from Hamiltonian Cycle in which the edit cost of a labeling is $n+m-2k$ when it realizes a longest path with $k$ edges; the optimal cost is $m-n+2pp(G)$, bounded in polynomial time by the matching number. We prove that irregularity alone forces $γ^{+}(G)\ge nΔ^{*}/(2m)-1$, where $Δ^{*}$ is the least $d\geΔ$ with $nd$ even, computable in linear time from the degree sequence; we characterize equality exactly. It is attained on the star, where $γ^{+}(K_{1,q})=(q-1)/2$ and the star maximizes $γ^{+}$, while $γ_{\triangle}$ stays bounded by an absolute constant. We determine paths and grids exactly, $γ^{+}(P_n)=γ^{+}(P_n\,\square\,P_n)=1/(n-1)$, and show $γ_{\triangle}(K_{1,q})\to 2$, not the $3/2$ suggested by the additive case. We report an exhaustive certified census of all $995$ connected graphs on at most seven vertices. The degree bound is attained on $89.4\%$ and the two invariants separate strictly on $84.7\%$, though both rates vary sharply with order: attainment $100\%,100\%,84.8\%,89.7\%$ and separation $0\%,61.9\%,73.2\%,87.7\%$ for $n=4,5,6,7$, dominated by the $853$ graphs on seven vertices. The star uniquely maximizes both. Edit count and the bi-Lipschitz distortion of the completed host are independent, moving oppositely on stars and paths.Data and certificates at doi:10.5281/zenodo.21852006.
