Structure Polynomials and Subgraphs of Rooted Regular Trees

Abstract

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.