Perfect state transfer, graph products and equitable partitions

We describe new constructions of graphs which exhibit perfect state transfer on continuous-time quantum walks. Our constructions are based on variants of the double cones [BCMS09,ANOPRT10,ANOPRT09] and the Cartesian graph products (which includes the n-cube) [CDDEKL05]. Some of our results include: (1) If $G$ is a graph with perfect state transfer at time $t_{G}$, where t_{G}\Spec(G) \subseteq \ZZ\pi, and $H$ is a circulant with odd eigenvalues, their weak product $G \times H$ has perfect state transfer. Also, if $H$ is a regular graph with perfect state transfer at time $t_{H}$ and $G$ is a graph where t_{H}|V_{H}|\Spec(G) \subseteq 2\ZZ\pi, their lexicographic product $G[H]$ has perfect state transfer. (2) The double cone $\overline{K}_{2} + G$ on any connected graph $G$, has perfect state transfer if the weights of the cone edges are proportional to the Perron eigenvector of $G$. This generalizes results for double cone on regular graphs studied in [BCMS09,ANOPRT10,ANOPRT09]. (3) For an infinite family \GG of regular graphs, there is a circulant connection so the graph K_{1}+\GG\circ\GG+K_{1} has perfect state transfer. In contrast, no perfect state transfer exists if a complete bipartite connection is used (even in the presence of weights) [ANOPRT09]. We also describe a generalization of the path collapsing argument [CCDFGS03,CDDEKL05], which reduces questions about perfect state transfer to simpler (weighted) multigraphs, for graphs with equitable distance partitions.Comment: 18 pages, 6 figure

Tensor and Matrix Inversions with Applications

Higher order tensor inversion is possible for even order. We have shown that a tensor group endowed with the Einstein (contracted) product is isomorphic to the general linear group of degree $n$. With the isomorphic group structures, we derived new tensor decompositions which we have shown to be related to the well-known canonical polyadic decomposition and multilinear SVD. Moreover, within this group structure framework, multilinear systems are derived, specifically, for solving high dimensional PDEs and large discrete quantum models. We also address multilinear systems which do not fit the framework in the least-squares sense, that is, when the tensor has an odd number of modes or when the tensor has distinct dimensions in each modes. With the notion of tensor inversion, multilinear systems are solvable. Numerically we solve multilinear systems using iterative techniques, namely biconjugate gradient and Jacobi methods in tensor format

From Data to Knowledge Graphs: A Multi-Layered Method to Model User's Visual Analytics Workflow for Analytical Purposes

The importance of knowledge generation drives much of Visual Analytics (VA). User-tracking and behavior graphs have shown the value of understanding users' knowledge generation while performing VA workflows. Works in theoretical models, ontologies, and provenance analysis have greatly described means to structure and understand the connection between knowledge generation and VA workflows. Yet, two concepts are typically intermixed: the temporal aspect, which indicates sequences of events, and the atemporal aspect, which indicates the workflow state space. In works where these concepts are separated, they do not discuss how to analyze the recorded user's knowledge gathering process when compared to the VA workflow itself. This paper presents Visual Analytic Knowledge Graph (VAKG), a conceptual framework that generalizes existing knowledge models and ontologies by focusing on how humans relate to computer processes temporally and how it relates to the workflow's state space. Our proposal structures this relationship as a 4-way temporal knowledge graph with specific emphasis on modeling the human and computer aspect of VA as separate but interconnected graphs for, among others, analytical purposes. We compare VAKG with relevant literature to show that VAKG's contribution allows VA applications to use it as a provenance model and a state space graph, allowing for analytics of domain-specific processes, usage patterns, and users' knowledge gain performance. We also interviewed two domain experts to check, in the wild, whether real practice and our contributions are aligned.Comment: 9 pgs, submitted to VIS 202