Average Steps until Absorption on Random Walks on Sea Dragon Trees

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.