发表机构
Eindhoven University of Technology; Vrije Universiteit Brussel; KU Leuven; Ghent University; Stichting Leerplanontwikkeling（埃因霍温理工大学; 布鲁塞尔自由大学; 荷语鲁汶大学; 根特大学; 学习计划开发基金会）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出基于有限环的统一构造框架，统一现有结果并产生三个含无穷多个严格 Neumaier 图的新族，刻画强正则条件，并证明分圆数阶至多七时四种环足以生成所有此类图。
AI 中文摘要
Neumaier 图是一种包含正则团（regular clique）的非完全边正则图（edge-regular graph）；若它不是强正则图（strongly regular graph），则称为严格 Neumaier 图（strictly Neumaier graph）。本文提出了一种利用有限环（finite rings）的构造方法，该方法统一了若干已知结果，并产生了三个新族，每个族均包含无穷多个严格 Neumaier 图。我们刻画了所得图何时为强正则图。由有限域扩张的范数函数（norm functions）得到的族，包含了由经典超椭圆（classical hyperoval）导出的 *-Tits 型广义四边形（generalized quadrangles）的共线图（collinearity graphs）。我们进一步证明，当所涉及的分圆数（cyclotomic numbers）的阶至多为七时，四种类型的环足以在同构意义下获得由我们的构造产生的所有严格 Neumaier 图。作为我们结果的一个推论，这四种类型中的每一种现在都已被用于构造 Neumaier 图。
英文摘要
A Neumaier graph is a non-complete edge-regular graph containing a regular clique; it is called strictly Neumaier if it is not strongly regular. In this paper we present a construction using finite rings that unifies several known results and yields three new families, each containing infinitely many strictly Neumaier graphs. We characterize when the resulting graphs are strongly regular. The family obtained from norm functions of finite field extensions includes the collinearity graphs of generalized quadrangles of \(*\)-Tits type arising from the classical hyperoval. We further show that, when the cyclotomic numbers involved have order at most seven, four types of rings suffice to obtain, up to isomorphism, all strictly Neumaier graphs arising from our construction. As a consequence of our results, each of these four types has now been used to construct Neumaier graphs.
