Extensions of Barrier Sets to Nonzero Roots of the Matching Polynomials

In matching theory, barrier sets (also known as Tutte sets) have been studied extensively due to its connection to maximum matchings in a graph. In this paper, we first define $\theta$-barrier sets. Our definition of a $\theta$-barrier set is slightly different from that of a barrier set. However we show that $\theta$-barrier sets and barrier sets have similar properties. In particular, we prove a generalized Berge's Formula and give a characterization for the set of all $\theta$-special vertices in a graph

Average mixing of continuous quantum walks

If $X$ is a graph with adjacency matrix $A$, then we define $H(t)$ to be the operator $\exp(itA)$. The Schur (or entrywise) product $H(t)\circ H(-t)$ is a doubly stochastic matrix and, because of work related to quantum computing, we are concerned the \textsl{average mixing matrix}. This can be defined as the limit of C^{-1} \int_0^C H(t)\circ H(-t)\dt as $C\to\infty$. We establish some of the basic properties of this matrix, showing that it is positive semidefinite and that its entries are always rational. We find that for paths and cycles this matrix takes on a surprisingly simple form, thus for the path it is a linear combination of $I$, $J$ (the all-ones matrix), and a permutation matrix.Comment: 20 pages, minor fixes, added section on discrete walks; fixed typo

When can perfect state transfer occur?

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 $|H({\tau})_{u,v}| = 1$. Our chief problem is to characterize the cases where perfect state transfer occurs. We show that if perfect state transfer does occur in a graph, then the spectral radius is an integer or a quadratic irrational; using this we prove that there are only finitely many graphs with perfect state transfer and with maximum valency at most 4K4. We also show that if perfect state transfer from $u$ to $v$ occurs, then the graphs $X\setminus u$ and $X\setminus v$ are cospectral and any automorphism of $X$ that fixes $u$ must fix $v$ (and conversely).Comment: 16 page

Bose-Mesner Algebras attached to Invertible Jones Pairs

In 1989, Vaughan Jones introduced spin models and showed that they could be used to form link invariants in two different ways--by constructing representations of the braid group, or by constructing partition functions. These spin models were subsequently generalized to so-called 4-weight spin models by Bannai and Bannai; these could be used to construct partition functions, but did not lead to braid group representations in any obvious way. Jaeger showed that spin models were intimately related to certain association schemes. Yamada gave a construction of a symmetric spin model on $4n$ vertices from each 4-weight spin model on $n$ vertices. In this paper we build on recent work with Munemasa to give a different proof to Yamada's result, and we analyse the structure of the association scheme attached to this spin model.Comment: 23 page