发表机构
Jadavpur University; University of Calcutta（贾达普大学; 加尔各答大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文将广义图合成应用于有限群差图研究，通过约简得到基于循环子群的图B(G)，完整刻画了所有有限群差图的非空性与连通性，并给出精确直径界。
AI 中文摘要
有限群G的差图D(G)由其交幂图与幂图的边差集经删除孤立顶点后得到，该图已被研究，且对满足特定条件的有限群给出了连通性的充分条件及直径界6。本文使用广义图合成将D(G)约简为基于G的循环子群的图B(G)，使连通性与直径由G的子群结构决定。本文依据分支子群与b-正规性得到B(G)非空的一般判据，刻画了有限p-群、非循环有限阿贝尔群、中心平凡与非平凡的非阿贝尔群的连通性，结合已确立的循环群情形，给出所有有限群差图非空性与连通性的完整刻画。各连续结构情形自然导出精确直径界2、3、4、5；对于无中心非阿贝尔群，连通性由唯一分支子群或辅助图A(G)决定，后者满足直径A(G)-1 ≤ 直径B(G) ≤ max{4,直径A(G)+1}，且两个界均为精确的。
英文摘要
The difference graph $D(G)$ of a finite group $G$ is obtained from the edge difference between its intersection power graph and power graph, after deleting isolated vertices. This graph has already been studied, with sufficient conditions for connectedness and a diameter bound $6$ for finite groups satisfying those conditions. We use generalized graph composition to reduce $D(G)$ to a graph $B(G)$ on the cyclic subgroups of $G$, so that connectedness and diameter are determined by the subgroup structure of $G$. We obtain a general criterion for the non-emptiness of $B(G)$ in terms of branching subgroups and b-normality, and characterize its connectedness for finite $p$-groups, non-cyclic finite abelian groups, and non-abelian groups with both trivial and non-trivial center. Combined with the previously established cyclic-group case, this gives a complete characterization of non-emptiness and connectedness of difference graphs for all finite groups. The successive structural cases lead naturally to the sharp diameter bounds $2,3,4,$ and $5$. For centerless non-abelian groups, connectedness is governed either by a unique branching subgroup or by an auxiliary graph $\mathcal A(G)$; in the latter case \[ \operatorname{diam}\mathcal A(G)-1 \leq \operatorname{diam}B(G) \leq \max\{4,\operatorname{diam}\mathcal A(G)+1\}, \] and both bounds are sharp.
Comments51 pages, 11 figures
