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Pretty good plus state transfer in cycles

Sarojini Mohapatra, Hiranmoy Pal

Abstract

We investigate fractional revival in graphs with respect to the adjacency, Laplacian, and signless Laplacian matrices. We observe that, under certain conditions, fractional revival is preserved under graph complementation. Then we establish a connection between fractional revival in a graph and in its double cover, and obtain a complete characterization of pretty good plus state transfer in cycles and their complements. This leads to characterizations of pretty good vertex state transfer in weighted paths with potential.

Pretty good plus state transfer in cycles

Abstract

We investigate fractional revival in graphs with respect to the adjacency, Laplacian, and signless Laplacian matrices. We observe that, under certain conditions, fractional revival is preserved under graph complementation. Then we establish a connection between fractional revival in a graph and in its double cover, and obtain a complete characterization of pretty good plus state transfer in cycles and their complements. This leads to characterizations of pretty good vertex state transfer in weighted paths with potential.
Paper Structure (6 sections, 26 theorems, 39 equations, 7 figures)

This paper contains 6 sections, 26 theorems, 39 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Algebraic properties
  3. State transfer in graph complement
  4. Double covers
  5. Plus PGST in cycles
  6. PGST in weighted paths with potential

Key Result

Theorem 1

Let a graph $G$ admit pretty good plus state transfer between $\frac{1}{\sqrt{2}}\left({\mathbf e}_a+{\mathbf e}_b\right)$ and $\frac{1}{\sqrt{2}}\left({\mathbf e}_c+{\mathbf e}_d\right).$ Then the graph $G$ admits pretty good pair state transfer if there exists an automorphism of $G$ with permutati

Figures (7)

  • Figure 1: $C_8$ as double cover of $C_4.$
  • Figure 2:
  • Figure 3:
  • Figure 4:
  • Figure 6:
  • ...and 2 more figures

Theorems & Definitions (45)

  • Remark 1
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Corollary 1
  • Example 1
  • Remark 2
  • ...and 35 more