Geodesic
Monotonicity of random walks' states on finite grids
Journal of the Belarusian State University. Mathematics and Informatics, Tome 1 (2022), pp. 38-45.

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In this paper two ways to order the nodes of a graph with respect to an arbitrary node are considered, both connected to random walks on the graph. The first one is the order according to probabilities of states of a random walk of fixed length started in that arbitrary node. The walks considered here are lazy walks – instead of making a step they are allowed to stay in the same node. A class of graphs, where such order the corresponds to the weak order by geodesic distances, was found. Square and toric $n$-dimensional grids are shown to be instances of this class. The second way of ordering is resistance distance to a fixed node. For another class of graphs, a pair of vertices with maximal resistance distance between them is established. Grids are again shown to be an example of graphs belonging to this class.
Keywords: random walks; resistance distance; grids. 
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A. O. Zadorozhnuyk. Monotonicity of random walks' states on finite grids. Journal of the Belarusian State University. Mathematics and Informatics, Tome 1 (2022), pp. 38-45. http://geodesic.mathdoc.fr/item/BGUMI_2022_1_a4/

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