Average Steps until Absorption on Random Walks on Sea Dragon Trees

TL;DR

This paper studies the average steps until absorption in random walks on a special class of trees called sea dragons, using Markov chains to analyze various cases and related equations.

Contribution

It introduces the concept of sea dragons and provides methods to compute absorption times for these trees and general graphs.

Findings

Derived formulas for absorption times on sea dragons.

Analyzed equations related to average steps until absorption.

Extended results to broader classes of graphs.

Abstract

For a graph $G$ and vertices $u,v$, we define the ASUA of $v$, $t(G,v,u)$, to be the average steps until absorption along a random walk terminating at $u$. We define a sea dragon to be a tree with a unique path $P$ such that if $d(u) \geq 3$ for some vertex $u$, then $u \in V(P)$. We use Markov chains to determine $t(G,v,u)$ for all vertices of several classes of sea dragons, a broad subclass of trees. Additionally, we give several results on equations related to ASUAs on general graphs.