Single Polygon Counting for $m$ Fixed Nodes in Cayley Tree: Two Extremal Cases

TL;DR

This paper derives exact formulas for counting polygons with fixed nodes in Cayley trees, providing insights into their asymptotic behavior, especially in two extremal cases involving full components and fixed vertices.

Contribution

It introduces precise formulas for polygon counting in Cayley trees for two specific extremal cases, expanding combinatorial understanding.

Findings

Formulas expressed as linear combinations of Catalan numbers

Asymptotic estimates derived from the formulas

Exact counts for polygons containing full components or fixed vertices

Abstract

We denote a polygon as a connected component in Cayley tree of order 2 containing certain number of fix vertices. We found an exact formula for a polygon counting problem for two cases, in which, for the first case the polygon contain a full connected component of a Cayley tree and for the second case the polygon contain two fixed vertices. From these formulas, which is in the form of finite linear combination of Catalan numbers, one can find the asymptotic estimation for a counting problem.