AI 中文总结
本研究探讨了通过同态计数区分具有简单谱的图，证明了此类图类需满足无界树宽等条件，并构造了相关图对，扩展了三次同态计数的应用范围。
AI 中文摘要
我们研究哪些图类通过其成员的同态计数能确定每个具有简单谱的图（在同构意义下）。我们证明这样的图类必须具有无界树宽和无界欧拉亏格，并且包含一个以七顶点完全图作为子式的图。为证明这些条件，我们构造了非 isomorphic 且具有简单谱的共谱图对，每对图都是二分图、非奇异且最大度至多六。该构造将根树附加到 Cai-Fürer-Immerman 图上，并保持对每个 minor-closed 类的同态不可区分性。然后我们证明三次同态计数确定每个其邻接矩阵核维数至多为一的图。这推广了非奇异图的已知结果，并适用于所有具有简单谱的图。我们还证明两个图具有相同的三次同态计数当且仅当它们具有相同的顶点数，并且存在一个双射线性等距映射，在它们邻接矩阵的像之间交织邻接算子并保持逐坐标三线性形式。最后，我们证明来自单顶点图和来自任何连通三次图序列的细分图的同态计数，与三次同态计数具有相同的区分能力，如果每个完全图都是该序列中某个成员的子式。
英文摘要
We study which graph classes determine every graph with simple spectrum up to isomorphism by homomorphism counts from their members. We show that such a class has unbounded treewidth and Euler genus and contains a graph with the complete graph on seven vertices as a minor. To prove these conditions, we construct pairs of non-isomorphic cospectral graphs with simple spectrum, each bipartite, nonsingular and of maximum degree at most six. The construction attaches rooted trees to Cai-Furer-Immerman graphs and preserves homomorphism indistinguishability over every minor-closed class. We then prove that subcubic homomorphism counts determine every graph whose adjacency matrix has kernel dimension at most one. This extends the known result for nonsingular graphs and applies to all graphs with simple spectrum. We also prove that two graphs have the same subcubic homomorphism counts if and only if they have the same number of vertices and there is a bijective linear isometry between the images of their adjacency matrices that intertwines the adjacency operators and preserves the coordinatewise trilinear forms. Finally, we prove that homomorphism counts from the single-vertex graph and from subdivisions of any sequence of connected cubic graphs have the same distinguishing power as subcubic homomorphism counts if every complete graph is a minor of some member of the sequence.
Comments29 pages, 8 figures
