Sedentary quantum walks on bipartite graphs

TL;DR

This paper investigates the phenomenon of vertex sedentariness in quantum walks on bipartite graphs, revealing conditions under which vertices tend to stay at home and contrasting behaviors across different graph classes.

Contribution

It establishes that most planar graphs and trees have at least two sedentary vertices, while bipartite graphs with certain spectral properties have no sedentary vertices, and introduces new graph constructions related to sedentariness.

Findings

Most planar graphs and trees have at least two sedentary vertices.

Vertices in nonsingular weighted bipartite graphs are not sedentary.

Unweighted paths and even cycles contain no sedentary vertices.

Abstract

If a quantum walk starting on a vertex tends to stay at home, then that vertex is said to be sedentary. We prove that almost all planar graphs and almost all trees contain at least two sedentary vertices for any assignment of edge weights -- a result that suggests vertex sedentariness is a common phenomenon in trees and planar graphs. For weighted bipartite graphs, we show that a vertex is not sedentary whenever 0 does not belong to its eigenvalue support. Consequently, each vertex in a nonsingular weighted bipartite graph is not sedentary, a stark contrast to weighted trees and weighted planar graphs. A corollary of this result is that every vertex in a bipartite graph with a unique perfect matching is not sedentary for any assignment of edge weights. We also construct new families of weighted bipartite graphs with sedentary vertices using the bipartite double and subdivision operations. Finally, we show that unweighted paths and unweighted even cycles contain no sedentary vertices.