摘要
We introduce an algebraic structure allowing us to describe subgraphs of a regular rooted tree. Its elements are called structure polynomials, and they are in a one- to-one correspondence with the set of all subgraphs of the tree. We define two operations, the sum and the product of structure polynomials, giving a graph interpretation of them. Then we introduce an equivalence relation between polynomials, using the action of the full automorphism group of the tree, and we count equivalence classes of subgraphs modulo this equivalence. We also prove that this action gives rise to symmetric Gelfand pairs. Finally, when the regularity degree of the tree is a prime p, we regard each level of the tree as a finite dimensional vector space over the finite field Fp, and we are able to completely characterize structure polynomials corresponding to subgraphs whose leaf set is a vector subspace.
我们介绍允许我们描述一棵常规生根的树的 subgraphs 的代数学的结构。它的元素被称为结构多项式，并且他们在 -- 与树的所有 subgraphs 的集合的一个通讯。我们定义二操作，和和结构多项式的产品，给他们的图解释。然后，我们介绍在多项式之间的一种等价关系，用树的完整的自守组的行动，并且我们数 subgraphs 模的等价班这个等价。我们也证明这个行动产生对称的 Gelfand 对。最后，当树的整齐度是主要 p 时，我们在有限的地 𝔽 上认为树的每水平是一个有限维的向量空格； p，和我们能完全描绘相应于其叶集合是向量 subspace 的 subgraphs 的结构多项式。
