261 research outputs found
Subclasses of Normal Helly Circular-Arc Graphs
A Helly circular-arc model M = (C,A) is a circle C together with a Helly
family \A of arcs of C. If no arc is contained in any other, then M is a proper
Helly circular-arc model, if every arc has the same length, then M is a unit
Helly circular-arc model, and if there are no two arcs covering the circle,
then M is a normal Helly circular-arc model. A Helly (resp. proper Helly, unit
Helly, normal Helly) circular-arc graph is the intersection graph of the arcs
of a Helly (resp. proper Helly, unit Helly, normal Helly) circular-arc model.
In this article we study these subclasses of Helly circular-arc graphs. We show
natural generalizations of several properties of (proper) interval graphs that
hold for some of these Helly circular-arc subclasses. Next, we describe
characterizations for the subclasses of Helly circular-arc graphs, including
forbidden induced subgraphs characterizations. These characterizations lead to
efficient algorithms for recognizing graphs within these classes. Finally, we
show how do these classes of graphs relate with straight and round digraphs.Comment: 39 pages, 13 figures. A previous version of the paper (entitled
Proper Helly Circular-Arc Graphs) appeared at WG'0
Spectral and wave function statistics in Quantum digraphs
This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.Spectral and wave function statistics of the quantum directed graph, QdG, are studied. The crucial feature of this model is that the direction of a bond (arc) corresponds to the direction of the waves propagating along it. We pay special attention to the full Neumann digraph, FNdG, which consists of pairs of antiparallel arcs between every node, and differs from the full Neumann graph, FNG, in that the two arcs have two incommensurate lengths. The spectral statistics of the FNG (with incommensurate bond lengths) is believed to be universal, i.e. to agree with that of the random matrix theory, RMT, in the limit of large graph size. However, the standard perturbative treatment of the field theoretical representation of the 2-point correlation function [1, 2] for a FNG, does not account for this behaviour. The nearest-neighbor spacing distribution of the closely related FNdG is studied numerically. An original, efficient algorithm for the generation of the spectrum of large graphs allows for the observation that the distribution approaches indeed universality at increasing graph size (although the convergence cannot be ascertained), in particular "level repulsion" is confirmed. The numerical technique employs a new secular equation which generalizes the analogous object known for undirected graphs [3, 4], and is based on an adaptation to digraphs of the idea of wave function continuity. In view of the contradiction between the field theory [2] and the strong indications of universality, a non-perturbative approach to analysing the universal limit is presented. The substitution of the FNG by the FNdG results in a field theory with fewer degrees of freedom. Despite this simplification, the attempt is inconclusive. Possible applications of this approach are suggested.
Regarding the wave function statistics, a field theoretical representation for the
spectral average of the wave intensity on an fixed arc is derived and studied in the universal limit. The procedure originates from the study of wave function statistics on disordered metallic grains [5] and is used in conjunction with the field theory approach pioneered in [2]
GiViP: A Visual Profiler for Distributed Graph Processing Systems
Analyzing large-scale graphs provides valuable insights in different
application scenarios. While many graph processing systems working on top of
distributed infrastructures have been proposed to deal with big graphs, the
tasks of profiling and debugging their massive computations remain time
consuming and error-prone. This paper presents GiViP, a visual profiler for
distributed graph processing systems based on a Pregel-like computation model.
GiViP captures the huge amount of messages exchanged throughout a computation
and provides an interactive user interface for the visual analysis of the
collected data. We show how to take advantage of GiViP to detect anomalies
related to the computation and to the infrastructure, such as slow computing
units and anomalous message patterns.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
Obstruction characterization of co-TT graphs
Threshold tolerance graphs and their complement graphs ( known as co-TT
graphs) were introduced by Monma, Reed and Trotter[24]. Introducing the concept
of negative interval Hell et al.[19] defined signed-interval bigraphs/digraphs
and have shown that they are equivalent to several seemingly different classes
of bigraphs/digraphs. They have also shown that co-TT graphs are equivalent to
symmetric signed-interval digraphs. In this paper we characterize
signed-interval bigraphs and signed-interval graphs respectively in terms of
their biadjacency matrices and adjacency matrices. Finally, based on the
geometric representation of signed-interval graphs we have setteled the open
problem of forbidden induced subgraph characterization of co-TT graphs posed by
Monma, Reed and Trotter in the same paper.Comment: arXiv admin note: substantial text overlap with arXiv:2206.0591
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