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New results in vertex sedentariness

Hermie Monterde

Abstract

A vertex in a graph is said to be sedentary if a quantum state assigned on that vertex tends to stay on that vertex. Under mild conditions, we show that the direct product and join operations preserve vertex sedentariness. We also completely characterize sedentariness in blow-up graphs. These results allow us to construct new infinite families of graphs with sedentary vertices. We prove that a vertex with a twin is either sedentary or admits pretty good state transfer. Moreover, we give a complete characterization of twin vertices that are sedentary, and provide sharp bounds on their sedentariness. As an application, we determine the conditions in which perfect state transfer, pretty good state transfer and sedentariness occur in complete bipartite graphs and threshold graphs of any order.

New results in vertex sedentariness

Abstract

A vertex in a graph is said to be sedentary if a quantum state assigned on that vertex tends to stay on that vertex. Under mild conditions, we show that the direct product and join operations preserve vertex sedentariness. We also completely characterize sedentariness in blow-up graphs. These results allow us to construct new infinite families of graphs with sedentary vertices. We prove that a vertex with a twin is either sedentary or admits pretty good state transfer. Moreover, we give a complete characterization of twin vertices that are sedentary, and provide sharp bounds on their sedentariness. As an application, we determine the conditions in which perfect state transfer, pretty good state transfer and sedentariness occur in complete bipartite graphs and threshold graphs of any order.
Paper Structure (10 sections, 30 theorems, 24 equations, 1 figure)

This paper contains 10 sections, 30 theorems, 24 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Sedentariness
  4. Graphs with twins
  5. Complete multipartite graphs
  6. Threshold graphs
  7. Direct Product
  8. Blow-ups
  9. Joins
  10. Open questions

Key Result

Proposition 2

A sedentary vertex cannot be involved in pretty good state transfer. Conversely, a vertex involved in pretty good state transfer cannot be sedentary.

Figures (1)

  • Figure 1: The graph $P_n'$ with twin vertices $u$ and $v$ marked white

Theorems & Definitions (62)

  • Definition 1
  • Proposition 2
  • Example 3
  • Theorem 4
  • Remark 5
  • Lemma 6
  • Remark 7
  • Lemma 8
  • Lemma 9
  • Lemma 10
  • ...and 52 more