The main eigenvalues of a graph: a survey

We survey results relating main eigenvalues and main angles to the structure of a graph. We provide a number of short proofs, and note the connection with star partitions. We discuss graphs with just two main eigenvalues in the context of measures of irregularity, and in the context of harmonic graphs

A theory of spectral partitions of metric graphs

We introduce an abstract framework for the study of clustering in metric graphs: after suitably metrising the space of graph partitions, we restrict Laplacians to the clusters thus arising and use their spectral gaps to define several notions of partition energies; this is the graph counterpart of the well-known theory of spectral minimal partitions on planar domains and includes the setting in [Band \textit{et al}, Comm.\ Math.\ Phys.\ \textbf{311} (2012), 815--838] as a special case. We focus on the existence of optimisers for a large class of functionals defined on such partitions, but also study their qualitative properties, including stability, regularity, and parameter dependence. We also discuss in detail their interplay with the theory of nodal partitions. Unlike in the case of domains, the one-dimensional setting of metric graphs allows for explicit computation and analytic -- rather than numerical -- results. Not only do we recover the main assertions in the theory of spectral minimal partitions on domains, as studied in [Conti \textit{et al}, Calc.\ Var.\ \textbf{22} (2005), 45--72; Helffer \textit{et al}, Ann.\ Inst.\ Henri Poincar\'e Anal.\ Non Lin\'eaire \textbf{26} (2009), 101--138], but we can also generalise some of them and answer (the graph counterparts of) a few open questions

SMART-KG: Hybrid Shipping for SPARQL Querying on the Web

While Linked Data (LD) provides standards for publishing (RDF) and (SPARQL) querying Knowledge Graphs (KGs) on the Web, serving, accessing and processing such open, decentralized KGs is often practically impossible, as query timeouts on publicly available SPARQL endpoints show. Alternative solutions such as Triple Pattern Fragments (TPF) attempt to tackle the problem of availability by pushing query processing workload to the client side, but suffer from unnecessary transfer of irrelevant data on complex queries with large intermediate results. In this paper we present smart-KG, a novel approach to share the load between servers and clients, while significantly reducing data transfer volume, by combining TPF with shipping compressed KG partitions. Our evaluations show that outperforms state-of-the-art client-side solutions and increases server-side availability towards more cost-effective and balanced hosting of open and decentralized KGs.Series: Working Papers on Information Systems, Information Business and Operation

Quantum ergodicity for graphs related to interval maps

We prove quantum ergodicity for a family of graphs that are obtained from ergodic one-dimensional maps of an interval using a procedure introduced by Pakonski et al (J. Phys. A, v. 34, 9303-9317 (2001)). As observables we take the L^2 functions on the interval. The proof is based on the periodic orbit expansion of a majorant of the quantum variance. Specifically, given a one-dimensional, Lebesgue-measure-preserving map of an interval, we consider an increasingly refined sequence of partitions of the interval. To this sequence we associate a sequence of graphs, whose directed edges correspond to elements of the partitions and on which the classical dynamics approximates the Perron-Frobenius operator corresponding to the map. We show that, except possibly for subsequences of density 0, the eigenstates of the quantum graphs equidistribute in the limit of large graphs. For a smaller class of observables we also show that the Egorov property, a correspondence between classical and quantum evolution in the semiclassical limit, holds for the quantum graphs in question.Comment: 20 pages, 1 figur

Outer space for untwisted automorphisms of right-angled Artin groups

For a right-angled Artin group $A_\Gamma$, the untwisted outer automorphism group $U(A_\Gamma)$ is the subgroup of $Out(A_\Gamma)$ generated by all of the Laurence-Servatius generators except twists (where a {\em twist} is an automorphisms of the form $v\mapsto vw$ with $vw=wv$). We define a space $\Sigma_\Gamma$ on which $U(A_\Gamma)$ acts properly and prove that $\Sigma_\Gamma$ is contractible, providing a geometric model for $U(A_\Gamma)$ and its subgroups. We also propose a geometric model for all of $Out(A_\Gamma)$ defined by allowing more general markings and metrics on points of $\Sigma_\Gamma$.Comment: Example section added, introduction modified. Final version, to appear in Geometry and Topolog