Asymptotic behavior of the transition probability of a simple random walk on a line graph

Abstract

For simple random walks $\left\{P_G^n\right\}$ on a homogeneous graph $G$ and $\left\{P_{L\left(G\right)}^n\right\}$ on its line graph $L\left(G\right)$, we obtain the relationship between the asymptotic behavior of the $n$-step transition probability $P_G^n(x,x)$ and that of $P_{L\left(G\right)}^n(x,x)$ as $n\to \infty$.