具有强正则子图的Deza图的无穷级数

中文摘要 AI 辅助

若图Γ具有n个顶点、k-正则性，且任意两个不同顶点u和v的公共邻接点数目仅为a或b，则称其为参数为(n,k,b,a)的Deza图。以Γ的顶点集作为顶点集，且两顶点相邻当且仅当它们有a个或b个公共邻接点的图Δ₁和Δ₂，被称为该Deza图的子图。若Deza图Γ的Δ₁和Δ₂均为强正则图，则称Γ为强Deza图。本研究构造了无穷族强Deza图，其子图Δ₁和Δ₂是与图NO^{ε⊥}_n(5)及其补图overline{NO^{ε⊥}_n(5)}具有相同参数的强正则图。

英文摘要

A Deza graph is a regular graph in which the number of common neighbours of two distinct vertices takes at most two values, regardless of adjacency. Its children are the graphs on the same vertex set in which adjacency is determined by these two common-neighbour counts. A Deza graph is called strongly Deza if both children are strongly regular. We construct an infinite family of edge-regular strongly Deza graphs using non-degenerate quadratic forms over the field with five elements. For every odd dimension greater than three and each of the two determinant square classes, we obtain a graph on the projective points represented by vectors of norm one. Its children are complementary strongly regular graphs with the parameters of the corresponding orthogonality graph on non-isotropic points and its complement. We determine the parameters by counting solutions to systems involving the associated bilinear form. These counts also yield symmetric association schemes over finite fields of odd characteristic. Further constructions include Deza graphs in dimension four, orthogonality graphs in odd dimensions, unions of relations in even dimensions over the field with nine elements, and an odd-dimensional family over the field with thirteen elements. We also give low-dimensional examples, including one whose children are a triangular graph and its complement.

发表机构

• Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences（俄罗斯科学院乌拉尔分校克拉索夫斯基数学与力学研究所）
• Ural Mathematical Center（乌拉尔数学中心）
• Ural Federal University named after the First President of Russia B. N. Yeltsin（俄罗斯首任总统B.N.叶利钦乌拉尔联邦大学）

机构由 AI 辅助整理，请以论文原文为准。

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23 pages