Structured eigenbases and pair state transfer on threshold graphs

TL;DR

This work leverages the known Laplacian eigenbasis shared by stars and threshold graphs to address two main goals: (i) fully characterize threshold graphs with simply structured Laplacian eigenspaces (basis entries in ${-1,0,1}$) and establish sufficient conditions for these graphs to be WHD, with a complete enumeration up to $n\le 20$ vertices; and (ii) characterize Laplacian pair state transfer on threshold graphs, showing that vertex state transfer and pair state transfer are equivalent unless the graph is a join of a complete and empty graph in certain configurations, and identifying infinite families where pair PST occurs without vertex PST. The paper provides a precise description in terms of the binary sequence blocks $(s_i,t_i)$, establishes modular conditions for WHD, and gives a thorough analysis of strong cospectrality and spectral supports governing pair-state quantum transport on threshold graphs. These results advance spectral graph theory for threshold graphs, contribute to the study of quantum walks on cographs, and offer concrete tools for constructing WHD graphs with controlled quantum transport properties.

Abstract

Recently, Macharete, Del-Vecchio, Teixeira and de Lima showed that a star and any threshold graph on the same number of vertices share the same eigenbasis relative to the Laplacian matrix. We use this fact to establish two main results in this paper. The first one is a characterization of threshold graphs that are \textit{simply structured}, i.e., their associated Laplacian matrices have eigenbases consisting of vectors with entries from the set $\{-1,0,1\}$. Then, we provide sufficient conditions such that a simply structured threshold graph is weakly Hadamard diagonalizable (WHD). This allows us to list all connected simply structured threshold graphs on at most 20 vertices, and identify those that are WHD. Second, we characterize Laplacian pair state transfer on threshold graphs. In particular, we show that the existence of Laplacian vertex state transfer and Laplacian pair state transfer on a threshold graph are equivalent if and only if it is not a join of a complete graph and an empty graph of certain sizes.

Figures (1)

• Figure 1: The structure of a threshold graph with $\mathbf{b}=\mathbf{0}^{s_1}\mathbf{1}^{t_1}\cdots \mathbf{0}^{s_r}\mathbf{1}^{t_r}$. Each vertex in $U_i$ is adjacent to $V_i \sqcup \cdots \sqcup V_r$, $V_1 \sqcup \cdots \sqcup V_r$ is a clique and $U_1 \sqcup \cdots \sqcup U_r$ is a coclique.

Theorems & Definitions (28)

• Definition 2.1
• Theorem 2.2
• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Lemma 3.3
• proof
• Theorem 3.4
• proof