强正则图、强多项式序列与两级多项式
Strongly regular graph, strongly polynomial sequences, and two-level polynomials
浏览论文内容
中文总结 AI 辅助
本文研究强正则图与强多项式图序列的特征多项式和色多项式的深度性质，证明强正则图特征多项式具有无限深度，而Paley图色多项式深度为二，并构造了使同态计数保持无限深度的目标序列。
中文摘要 AI 辅助
我们研究某些图族的特征多项式和色多项式的行为及其与同态计数的关系，重点关注每个固定余次数上系数对图族参数的多项式依赖性。我们证明了具有多项式参数的强正则图的特征多项式构成 Bogart 和 Woods 意义下的无限深度的两级多项式。相比之下，Paley 图的色多项式具有最大常数深度二：其三次余系数的系数并非最终是顶点数的多项式。我们还证明了每个在 de La Harpe 和 Jaeger 意义下的强多项式图序列的色多项式和特征多项式都具有无限深度。最后，我们研究两个强多项式图序列之间的同态计数。我们表明无限深度在一般情况下不一定成立，但我们构造了目标序列，使得对于每个强多项式源序列，无限深度都成立。
英文摘要
We study the behavior of the characteristic and chromatic polynomials of certain families of graphs and their relation to homomorphism counts, focusing on polynomial dependence of the coefficients at each fixed codegree on the graph family parameter. We prove that the characteristic polynomials of strongly regular graphs with polynomial parameters form a two-level polynomial of infinite depth in the sense of Bogart and Woods. In contrast, the chromatic polynomials of Paley graphs have maximal constant depth two: their coefficient of codegree three is not eventually polynomial in the number of vertices. We also prove that both the chromatic and characteristic polynomials of every strongly polynomial graph sequence, in the sense of de La Harpe and Jaeger, have infinite depth. Finally, we investigate homomorphism counts between two strongly polynomial graph sequences. We show that infinite depth need not hold in general, but we construct target sequences for which it holds for every strongly polynomial source sequence.
