Periodic Graphs

TL;DR

This paper investigates the conditions under which perfect state transfer occurs in graphs, linking it to periodicity and eigenvalues, and introduces new graph families exhibiting perfect state transfer using Hadamard matrices.

Contribution

It establishes a connection between perfect state transfer and periodicity, characterizes eigenvalues for periodic graphs, and constructs new examples of graphs with perfect state transfer.

Findings

Perfect state transfer implies graph periodicity at involved vertices.

Regular graphs with ≥4 eigenvalues are periodic iff their eigenvalues are integers.

Constructed new graphs with perfect state transfer using Hadamard matrices.

Abstract

Let $X$ be a graph on $n$ vertices with with adjacency matrix $A$ and let $H(t)$ denote the matrix-valued function $\exp(iAt)$. If $u$ and $v$ are distinct vertices in $X$, we say perfect state transfer}from $u$ to $v$ occurs if there is a time $\tau$ such that $|H(\tau)_{u,v}|=1$. If $u\in V(X)$ and there is a time $\sg$ such that $|H(\sg)_{u,u}|=1$, we say $X$ is periodic at $u$ with period $\sg$. We show that if perfect state transfer from $u$ to $v$ occurs at time $\tau$, then $X$ is periodic at both $u$ and $v$ with period $2\tau$. We extend previous work by showing that a regular graph with at least four distinct eigenvalues is periodic with respect to some vertex if and only if its eigenvalues are integers. We show that, for a class of graphs $X$ including all vertex-transitive graphs, if perfect state transfer occurs at time $\tau$, then $H(\tau)$ is a scalar multiple of a permutation matrix of order two with no fixed points. Using certain Hadamard matrices, we construct a new infinite family of graphs on which perfect state transfer occurs.