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Laplacian Pair State Transfer on Total Graphs

Akash Kalita, Bikash Bhattacharjya

TL;DR

The paper analyzes Laplacian pair state transfer on total graphs $\mathcal{T}(G)$ of $r$-regular graphs, establishing nonexistence of Laplacian PST for $r>2$ when $r+1$ is not a Laplacian eigenvalue and for complete graphs with more than three vertices. It then develops conditions under which Laplacian PGST occurs on $\mathcal{T}(G)$, leveraging spectral decompositions and Kronecker approximation to align phases, yielding infinitely many total graphs with PGST. The results distinguish non-bipartite and bipartite cases, providing concrete criteria such as $\frac{r+2}{4}$ being an integer and certain parity requirements. Together, these findings advance understanding of quantum state transfer on total graphs and identify broad families of graphs with controllable PST and PGST properties, including explicit examples like cocktail party graphs.

Abstract

The total graph of a graph $G$, denoted $\mathcal{T}(G)$, is defined as the graph whose vertex set is the union of the vertex set of $G$ and the edge set of $G$ such that two vertices of $\mathcal{T}(G)$ are adjacent if the corresponding elements of $G$ are adjacent or incident. In this paper, we investigate Laplacian perfect pair state transfer and Laplacian pretty good pair state transfer on $\mathcal{T}(G)$, where $G$ is an $r$-regular graph. We prove that if $r>2$ and $r+1$ is not a Laplacian eigenvalue of $G$, then $\mathcal{T}(G)$ fails to exhibit Laplacian perfect pair state transfer. We also prove that if $G$ is a complete graph on more than three vertices, then $\mathcal{T}(G)$ fails to exhibit Laplacian perfect pair state transfer. Further, we prove that under some mild conditions, $\mathcal{T}(G)$ exhibits Laplacian pretty good pair state transfer, where $G$ is an $r$-regular graph such that $r>2$ and $r+1$ is not a Laplacian eigenvalue of $G$. We use these conditions to obtain infinitely many total graphs exhibiting Laplacian pretty good pair state transfer.

Laplacian Pair State Transfer on Total Graphs

TL;DR

The paper analyzes Laplacian pair state transfer on total graphs of -regular graphs, establishing nonexistence of Laplacian PST for when is not a Laplacian eigenvalue and for complete graphs with more than three vertices. It then develops conditions under which Laplacian PGST occurs on , leveraging spectral decompositions and Kronecker approximation to align phases, yielding infinitely many total graphs with PGST. The results distinguish non-bipartite and bipartite cases, providing concrete criteria such as being an integer and certain parity requirements. Together, these findings advance understanding of quantum state transfer on total graphs and identify broad families of graphs with controllable PST and PGST properties, including explicit examples like cocktail party graphs.

Abstract

The total graph of a graph , denoted , is defined as the graph whose vertex set is the union of the vertex set of and the edge set of such that two vertices of are adjacent if the corresponding elements of are adjacent or incident. In this paper, we investigate Laplacian perfect pair state transfer and Laplacian pretty good pair state transfer on , where is an -regular graph. We prove that if and is not a Laplacian eigenvalue of , then fails to exhibit Laplacian perfect pair state transfer. We also prove that if is a complete graph on more than three vertices, then fails to exhibit Laplacian perfect pair state transfer. Further, we prove that under some mild conditions, exhibits Laplacian pretty good pair state transfer, where is an -regular graph such that and is not a Laplacian eigenvalue of . We use these conditions to obtain infinitely many total graphs exhibiting Laplacian pretty good pair state transfer.
Paper Structure (4 sections, 19 theorems, 55 equations)

This paper contains 4 sections, 19 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Basic Definitions and Preliminary Results
  3. Laplacian perfect pair state transfer
  4. Laplacian pretty good pair state transfer

Key Result

Theorem 2.1

Q. Chen Let $\textbf{e}_a - \textbf{e}_b$ and $\textbf{e}_c - \textbf{e}_d$ be two distinct pair states of a graph $G$. Also, let $\Phi_{ab} =\{\theta_0, \ldots, \theta_{\ell}\}$ such that $\theta_0 > \cdots > \theta_{\ell}$. Then $G$ exhibits Laplacian perfect pair state transfer between $\textbf{e If the conditions $(i)$, $(ii)$ and $(iii)$ hold, then $G$ exhibit Laplacian perfect pair state tra

Theorems & Definitions (31)

  • Theorem 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Lemma 2.4
  • Theorem 2.5
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • ...and 21 more