Quantum walks on blow-up graphs

Abstract

A blow-up of $n$ copies of a graph $G$ is the graph $\overset{n}\uplus~G$ obtained by replacing every vertex of $G$ by an independent set of size $n$, where the copies of vertices in $G$ are adjacent in the blow-up if and only if the vertices adjacent in $G$. Our goal is to investigate the existence of quantum state transfer on a blow-up graph $\overset{n}\uplus~G$, where the adjacency matrix is taken to be the time-independent Hamiltonian of the quantum system represented by $\overset{n}\uplus~G$. In particular, we establish necessary and sufficient conditions for vertices in a blow-up graph to exhibit strong cospectrality and various types of high probability quantum transport, such as periodicity, perfect state transfer (PST) and pretty good state transfer (PGST). It turns out, if $\overset{n}\uplus~G$ admits PST or PGST, then one must have $n=2.$ Moreover, if $G$ has an invertible adjacency matrix, then we show that every vertex in $\overset{2}\uplus~G$ pairs up with a unique vertex to exhibit strong cospectrality. We then apply our results to determine infinite families of graphs whose blow-ups admit PST and PGST.

Figures (2)

• Figure 1: The graph $\overset{2}{\uplus}~P_3$, a blow-up of two copies of $P_3$
• Figure 2: An eigenvector of $SK_{1,m}$ corresponding to $\pm1.$

Theorems & Definitions (38)

• Proposition 1
• Theorem 1
• Proposition 2
• Theorem 2
• proof
• Corollary 1
• Example 1
• Example 2
• Theorem 3
• proof