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Sedentary quantum walks on bipartite graphs

Karen Meagher, Hermie Monterde

TL;DR

The paper advances the theory of vertex sedentariness for continuous-time quantum walks on graphs, focusing on bipartite graphs, trees, and planar graphs. It develops spectral criteria and arithmetic tools showing that in weighted bipartite graphs, a vertex is not sedentary whenever its zero eigenvalue is absent from its eigenvalue support, implying nonsingular bipartite graphs have no sedentary vertices, and that almost all planar graphs and almost all trees contain sedentary vertices. It introduces constructive methods (bipartite double and subdivision) to create or avoid sedentary vertices and analyzes graphs with few eigenvalues to illustrate diverse behaviors. It also proves that unweighted paths and unweighted even cycles have no sedentary vertices, and provides a broad set of corollaries and open questions guiding future exploration of quantum transport on these graph families.

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.

Sedentary quantum walks on bipartite graphs

TL;DR

The paper advances the theory of vertex sedentariness for continuous-time quantum walks on graphs, focusing on bipartite graphs, trees, and planar graphs. It develops spectral criteria and arithmetic tools showing that in weighted bipartite graphs, a vertex is not sedentary whenever its zero eigenvalue is absent from its eigenvalue support, implying nonsingular bipartite graphs have no sedentary vertices, and that almost all planar graphs and almost all trees contain sedentary vertices. It introduces constructive methods (bipartite double and subdivision) to create or avoid sedentary vertices and analyzes graphs with few eigenvalues to illustrate diverse behaviors. It also proves that unweighted paths and unweighted even cycles have no sedentary vertices, and provides a broad set of corollaries and open questions guiding future exploration of quantum transport on these graph families.

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.
Paper Structure (11 sections, 46 theorems, 31 equations, 5 figures)

This paper contains 11 sections, 46 theorems, 31 equations, 5 figures.

Table of Contents

  1. Introduction
  2. State transfer
  3. Nonsingular bipartite graphs
  4. Singular bipartite graphs
  5. Trees
  6. Bipartite double
  7. Subdivision
  8. Bipartite graphs with few distinct eigenvalues
  9. Paths and cycles
  10. More constructions
  11. Future work

Key Result

Lemma 1

Let $W$ be a twin set in $X$. Then the vertices in $W$ are either all sedentary, or all involved in pretty good state transfer with each other. Moreover, if $|W|\geq 3$, then each vertex in $W$ is sedentary.

Figures (5)

  • Figure 1: The graph $G_n$ with pendent vertex $u$
  • Figure 2: The subdivided star $G(m)$ with vertices $u$ of degree $m$, $v$ of degree two and $w$ of degree one.
  • Figure 3: A graph $X$ with pendent vertices $u,v$ and $w$ sharing a common neighbor $x$.
  • Figure 4: The graph $G$ with vertex $v$ and a color class consisting of nonsedentary vertices marked white
  • Figure 5: The graphs $X$ (leftmost) and $X'$ (center left), which are weighted versions of $C_6$, and the graphs $Y$ (center right) and $Y'$ (rightmost), which are weighted versions of $C_8$

Theorems & Definitions (78)

  • Lemma 1
  • Theorem 2
  • Remark 3
  • Theorem 4
  • Definition 5
  • Lemma 6
  • Theorem 7
  • Corollary 8
  • proof
  • Lemma 9
  • ...and 68 more