We think of a random walk as an action of the time on the space. If the time is a digraph $G$ and the space is a digraph $H$, the random walk is represented by various statistics on the set of all homomorphisms from $G$ to $H.$ 

For two digraphs $G$ and $H$, let $hom (G,H)$ denote the number of homorphisms from $G$ to $H$.

Let $/mathcal A$ and $/mathcal B$ be two classes of digraphs. Each $G/in /mathcal A$ has a left-$/mathcal B$ homomorphism-profile, which is the vector $(hom (H,G))_{H/in /mathcal B}$ and a right-$/mathcal B$ homomorphism-profile, which is the vector $(hom (G,H))_{H/in /mathcal B}$. How can we compare two elements from $/mathcal A$ by their left- or right-$/mathcal B$ homomorphism-profiles?

We plan to introduce some relevant problems and conjectures, mainly about comparing different trees by their right-$/mathcal B$ homomorphism-profiles where $/mathcal B$ is the set of all digraphs or the set of all paths.

Joint work with Zeying Xu, Da Zhao and Yinfeng Zhu.

海报