Two particles' repelling random walks on the complete graph

Two particles ’ repelling random walks on the complete graph

We consider two particles’ repelling random walks on complete graphs. In this model, each particle has higher probability to visit the vertices which have been seldom visited by the other one. By a dynamical approach we prove that the two particles’ occupation measure asymptotically has small joint support almost surely if the repulsion is strong enough.

Collisions Among Random Walks on a Graph

17 There is one further consideration, which leads perhaps to the most intriguing conjecture of all. Let us put two tokens on a graph and let them take random walks, as before; but now suppose the schedule demon is clairvoyant|that is, he can see where each token will go, innnitely far into the future. The question is, with this advantage, can he now keep the tokens apart forever ? Of course we...

Restricted random walks on a graph

The problem of a restricted random walk on graphs which keeps track of the number of immediate reversal steps is considered by using a transfer matrix formulation. A closed-form expression is obtained for the generating function of the number of n-step walks with r reversal steps for walks on any graph. In the case of graphs of a uniform valence, we show that our result has a probabilistic mean...

Random Walks and Graph Homomorphisms

Graph homomorphisms through random walks

In this paper we introduce some general necessary conditions for the existence of graph homomorphisms, which hold in both directed and undirected cases. Our method is a combination of Diaconis and Saloff– Coste comparison technique for Markov chains and a generalization of Haemers interlacing theorem. As some applications, we obtain a necessary condition for the spanning subgraph problem, which...