一族正则整图的刻画与代数实现
Structural Characterizations and Algebraic Realizations of a Family of Regular Integral Graphs
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中文总结 AI 辅助
本文引入由群构造的新整图族，证明其谱唯一性，给出递归构造，并研究其作为有限2-群补图的代数实现条件。
中文摘要 AI 辅助
整图的所有特征值均为整数。整图极为罕见，在所有具有 $n$ 个顶点的图中，它们构成一个渐近消失的比例 $2^{-\Omega(n)}$。这使得构造新的整图族成为一项具有挑战性的任务。此外，大多数已知的无限整图族依赖于阿贝尔群上的凯莱图。在本文中，我们引入了一个由群构造的新整图族。我们由图构造的方式不同于凯莱图的构造。我们建立了一个谱唯一性定理，表明该无限族中的每个成员在所有有限简单图中由其邻接谱唯一确定。我们还给出了递归构造，从较小的成员生成较大的成员，从而提供了一类可扩展的整图。最后，我们研究了这些图作为有限 $2$-群的正素阶元素图的补图的代数实现，并获得了刻画此类实现的条件。我们还观察到，从不同非同构群获得的图具有共谱图。
英文摘要
All the eigenvalues of an integral graphs are integers. Integral graphs are extremely rare. They form an asymptotically vanishing fraction $2^{-Ω(n)}$ among all graphs on $n$ vertices. It makes the construction of a new family of integral graphs a challenging task. Also, most of the known infinite family of integral graphs rely on Cayley graphs over Abelian groups. In this article, we introduce a new family of integral graphs obtained from the groups. The construction of our graphs from groups is different from the construction of Cayley graphs. A spectral uniqueness theorem is established, which shows that each member of the infinite family is determined by its adjacency spectrum among all finite simple graphs. We also present recursive constructions that generates larger members of the family from smaller ones, providing a scalable class of integral graphs. Finally, we investigate algebraic realizations of these graphs as complements of Proper Prime Order Element Graphs of finite $2$-groups and obtain conditions characterizing such realizations. We also observe that the graphs obtained from different non-isomorphic groups have cospectral graphs.
发表机构
- National Institute of Technology Agartala（阿加塔拉国家技术学院）
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