Four-class Skew-symmetric Association Schemes

An association scheme is called skew-symmetric if it has no symmetric adjacency relations other than the diagonal one. In this paper, we study 4-class skew-symmetric association schemes. In J. Ma [On the nonexistence of skew-symmetric amorphous association schemes, submitted for publication], we discovered that their character tables fall into three types. We now determine their intersection matrices. We then determine the character tables and intersection numbers for 4-class skew-symmetric pseudocyclic association schemes, the only known examples of which are cyclotomic schemes. As a result, we answer a question raised by S. Y. Song [Commutative association schemes whose symmetrizations have two classes, J. Algebraic Combin. 5(1) 47-55, 1996]. We characterize and classify 4-class imprimitive skew-symmetric association schemes. We also prove that no 2-class Johnson scheme can admit a 4-class skew-symmetric fission scheme. Based on three types of character tables above, a short list of feasible parameters is generated.Comment: 12 page

Association schemes from the action of PGL(2,q)PGL(2,q) fixing a nonsingular conic in PG(2,q)

The group $PGL(2,q)$ has an embedding into $PGL(3,q)$ such that it acts as the group fixing a nonsingular conic in $PG(2,q)$. This action affords a coherent configuration $R(q)$ on the set $L(q)$ of non-tangent lines of the conic. We show that the relations can be described by using the cross-ratio. Our results imply that the restrictions $R_{+}(q)$ and $R_{-}(q)$ to the sets $L_{+}(q)$ of secant lines and to the set $L_{-}(q)$ of exterior lines, respectively, are both association schemes; moreover, we show that the elliptic scheme $R_{-}(q)$ is pseudocyclic. We further show that the coherent configuration $R(q^2)$ with $q$ even allow certain fusions. These provide a 4-class fusion of the hyperbolic scheme $R_{+}(q^2)$, and 3-class fusions and 2-class fusions (strongly regular graphs) of both schemes $R_{+}(q^2)$ and $R_{-}(q^2). The fusion results for the hyperbolic case are known, but our approach here as well as our results in the elliptic case are new.Comment: 33 page

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