We propose new families of graphs which exhibit quantum perfect state transfer. Our constructions are based on the join operator on graphs, its circulant generalizations, and the Cartesian product of graphs. We build upon the results of Ba\v{s}i\'{c} and Petkovi\'{c} ({\em Applied Mathematics Letters} {\bf 22}(10):1609-1615, 2009) and construct new integral circulants and regular graphs with perfect state transfer. More specifically, we show that the integral circulant $\textsc{ICG}_{n}(\{2,n/2^{b}\} \cup Q)$ has perfect state transfer, where $b \in \{1,2\}$, $n$ is a multiple of $16$ and $Q$ is a subset of the odd divisors of $n$. Using the standard join of graphs, we also show a family of double-cone graphs which are non-periodic but exhibit perfect state transfer. This class of graphs is constructed by simply taking the join of the empty two-vertex graph with a specific class of regular graphs. This answers a question posed by Godsil (arxiv.org math/08062074).

This paper studies quantum perfect state transfer on weighted graphs. We prove that the join of a weighted two-vertex graph with any regular graph has perfect state transfer. This generalizes a result of Casaccino et al. 1 where the regular graph is a complete graph with or without a missing edge. In contrast, we prove that the half-join of a weighted two-vertex graph with any weighted . Downloaded from www.worldscientific.com by UNIVERSITY OF VIRGINIA on 04/12/15. For personal use only. regular graph has no perfect state transfer. As a corollary, unlike for complete graphs, adding weights in complete bipartite graphs does not produce perfect state transfer. We also observe that any Hamming graph has perfect state transfer between each pair of its vertices. The result is a corollary of a closure property on weighted Cartesian products of perfect state transfer graphs. Moreover, on a hypercube, we show that perfect state transfer occurs between uniform superpositions on pairs of arbitrary subcubes, thus generalizing results of Bernasconi et al. 2 and Moore and Russell. 3

There are many people, including officemates, classmates and their spouses, roommates, neighbors, and the mathletes, who have made Iowa City feel like home for the past six years. In particular, I would like to thank Christine, Julia, Catie, Mario, Nathaniel, Colin, Kevin, and Ze for their friendship. I feel lucky to have been surrounded by such wonderful people. Finally, I am grateful to my mom, Nancy, my dad, Mike, my sister, Emily, and my partner, Nate, for their unwavering support and an earnest interest in my work. Without your encouragement, I might have given up a long time ago.

We propose new families of graphs which exhibit quantum perfect state transfer. Our constructions are based on the join operator on graphs, its circulant generalizations, and the Cartesian product of graphs. We build upon the results of Bašić et al. [5,4] and construct new integral circulants and regular graphs with perfect state transfer. More specifically, we show that the integral circulant ICG n ({2, n/2 b } ∪ Q) has perfect state transfer, where b ∈ {1, 2}, n is a multiple of 16 and Q is a subset of the odd divisors of n. Using the standard join of graphs, we also show a family of double-cone graphs which are non-periodic but exhibit perfect state transfer. This class of graphs is constructed by simply taking the join of the empty two-vertex graph with a specific class of regular graphs. This answers a question posed by Godsil [9].

We explore the relationship between two noncommutative generalizations of the classical Nevanlinna-Pick theorem: one proved by Constantinescu and Johnson in 2003 and the other proved by Muhly and Solel in 2004.