Twisted Duality, Cycle Family Graphs, and Embedded Graph Equivalence

2013 ◽  
pp. 43-60
Author(s):  
Joanna A. Ellis-Monaghan ◽  
Iain Moffatt
Keyword(s):  
Embedded Graph ◽  
Twisted Duality

SPECTRAL PROPERTIES AND SYMMETRIES OF DOUBLE COVERINGS OF GRAPHS WITH APPLICATIONS TO SUPERCONDUCTIVITY

2005 ◽  
Vol 15 (02) ◽  
pp. 301-324
Author(s):  
JACOB RUBINSTEIN ◽  
MICHELLE SCHATZMAN
Keyword(s):  
Magnetic Field ◽  
Phase Transition ◽  
Phase Transitions ◽  
Ground State ◽  
Spectral Properties ◽  
Embedded Graph ◽  
Symmetry Constraints ◽  
The Laplace Operator ◽  
The Magnetic Field ◽  
Double Coverings

Let M be a planar embedded graph, and let [Formula: see text] be its double covering. We count the multiplicity of the ground states of the Laplace operator on [Formula: see text] under certain symmetry constraints. The examples of interest for us are ladder-like graphs made out of n, identical rectangles. We find that in the case of an odd n, the multiplicity of the ground state is 2, and if n, is even, the ground state is simple. This result gives an answer to a conjecture by Parks on the type of phase transitions that can occur in a superconducting ladder: Parks conjectured that in the case when the magnetic field is one half fluxoid per rectangle, the phase transition would be continuous in the case of a ladder made out of two rectangles. Our result indeed implies Parks conjecture and generalizes it to any even ladder. The mathematics of this paper is a mixture of topology, symmetry arguments and comparison theorem between the eigenvalues of Laplace operators on graphs with well chosen boundary conditions.

scholarly journals Enumeration on graph mosaics

2017 ◽  
Vol 26 (05) ◽  
pp. 1750032 ◽  
Author(s):  
Kyungpyo Hong ◽  
Seungsang Oh
Keyword(s):  
Quantum System ◽  
Research Program ◽  
Knot Theory ◽  
Quantum Physics ◽  
Recursion Formula ◽  
Jones Polynomial ◽  
Exact Enumeration ◽  
State Matrix ◽  
Embedded Graph ◽  
Definition Of

Since the Jones polynomial was discovered, the connection between knot theory and quantum physics has been of great interest. Lomonaco and Kauffman introduced the knot mosaic system to give a definition of the quantum knot system that is intended to represent an actual physical quantum system. Recently the authors developed an algorithm producing the exact enumeration of knot mosaics, which uses a recursion formula of state matrices. As a sequel to this research program, we similarly define the (embedded) graph mosaic system by using 16 graph mosaic tiles, representing graph diagrams with vertices of valence 3 and 4. We extend the algorithm to produce the exact number of all graph mosaics. The magnified state matrix that is an extension of the state matrix is mainly used.

scholarly journals FROM PLANAR GRAPHS TO EMBEDDED GRAPHS - A NEW APPROACH TO KAUFFMAN AND VOGEL'S POLYNOMIAL

2000 ◽  
Vol 09 (08) ◽  
pp. 975-986 ◽  
Author(s):  
RUI PEDRO CARPENTIER
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Direct Proof ◽  
Connected Components ◽  
Embedded Graphs ◽  
New Approach ◽  
Graphical Calculus ◽  
Embedded Graph

In [4] Kauffman and Vogel constructed a rigid vertex regular isotopy invariant for unoriented four-valent graphs embedded in three dimensional space. It assigns to each embedded graph G a polynomial, denoted [G], in three variables, A, B and a, satisfying the skein relations: [Formula: see text] and is defined in terms of a state-sum and the Dubrovnik polynomial for links. Using the graphical calculus of [4] it is shown that the polynomial of a planar graph can be calculated recursively from that of planar graphs with less vertices, which also allows the polynomial of an embedded graph to be calculated without resorting to links. The same approach is used to give a direct proof of uniqueness of the (normalized) polynomial restricted to planar graphs. In the case B=A-1 and a=A, it is proved that for a planar graph G we have [G]=2c-1(-A-A-1)v, where c is the number of connected components of G and v is the number of vertices of G. As a corollary, a necessary, but not sufficient, condition is obtained for an embedded graph to be ambient isotopic to a planar graph. In an appendix it is shown that, given a polynomial for planar graphs satisfying the graphical calculus, and imposing the first skein relation above, the polynomial extends to a rigid vertex regular isotopy invariant for embedded graphs, satisfying the remaining skein relations. Thus, when existence of the planar polynomial is guaranteed, this provides a direct way, not depending on results for the Dubrovnik polynomial, to show consistency of the polynomial for embedded graphs.

scholarly journals Computing the Stretch of an Embedded Graph

2014 ◽  
Vol 28 (3) ◽  
pp. 1391-1401 ◽  
Author(s):  
Sergio Cabello ◽  
Markus Chimani ◽  
Petr Hliněný
Keyword(s):  
Embedded Graph

scholarly journals ANALYSIS OF SPATIO-TEMPORAL TRAFFIC PATTERNS BASED ON PEDESTRIAN TRAJECTORIES

2016 ◽  
Vol XLI-B2 ◽  
pp. 497-503
Author(s):  
S. Busch ◽  
T. Schindler ◽  
T. Klinger ◽  
C. Brenner
Keyword(s):  
Public Transportation ◽  
Autonomous Driving ◽  
Video Data ◽  
Observation Data ◽  
Traffic Patterns ◽  
Traffic Light ◽  
The Road ◽  
Embedded Graph ◽  
Observed Behaviour ◽  
Spatio Temporal

For driver assistance and autonomous driving systems, it is essential to predict the behaviour of other traffic participants. Usually, standard filter approaches are used to this end, however, in many cases, these are not sufficient. For example, pedestrians are able to change their speed or direction instantly. Also, there may be not enough observation data to determine the state of an object reliably, e.g. in case of occlusions. In those cases, it is very useful if a prior model exists, which suggests certain outcomes. For example, it is useful to know that pedestrians are usually crossing the road at a certain location and at certain times. This information can then be stored in a map which then can be used as a prior in scene analysis, or in practical terms to reduce the speed of a vehicle in advance in order to minimize critical situations. In this paper, we present an approach to derive such a spatio-temporal map automatically from the observed behaviour of traffic participants in everyday traffic situations. In our experiments, we use one stationary camera to observe a complex junction, where cars, public transportation and pedestrians interact. We concentrate on the pedestrians trajectories to map traffic patterns. In the first step, we extract trajectory segments from the video data. These segments are then clustered in order to derive a spatial model of the scene, in terms of a spatially embedded graph. In the second step, we analyse the temporal patterns of pedestrian movement on this graph. We are able to derive traffic light sequences as well as the timetables of nearby public transportation. To evaluate our approach, we used a 4 hour video sequence. We show that we are able to derive traffic light sequences as well as time tables of nearby public transportation.

scholarly journals On Self-Distilling Graph Neural Network

2021 ◽  
Author(s):  
Yuzhao Chen ◽  
Yatao Bian ◽  
Xi Xiao ◽  
Yu Rong ◽  
Tingyang Xu ◽  
...  
Keyword(s):  
Performance Enhancement ◽  
Training Process ◽  
Student Knowledge ◽  
Teacher Student ◽  
Embedded Graph ◽  
Training Cost ◽  
Knowledge Distillation ◽  
Discrepancy Rate ◽  
Standard Training ◽  
Graph Neural Networks

Recently, the teacher-student knowledge distillation framework has demonstrated its potential in training Graph Neural Networks (GNNs). However, due to the difficulty of training over-parameterized GNN models, one may not easily obtain a satisfactory teacher model for distillation. Furthermore, the inefficient training process of teacher-student knowledge distillation also impedes its applications in GNN models. In this paper, we propose the first teacher-free knowledge distillation method for GNNs, termed GNN Self-Distillation (GNN-SD), that serves as a drop-in replacement of the standard training process. The method is built upon the proposed neighborhood discrepancy rate (NDR), which quantifies the non-smoothness of the embedded graph in an efficient way. Based on this metric, we propose the adaptive discrepancy retaining (ADR) regularizer to empower the transferability of knowledge that maintains high neighborhood discrepancy across GNN layers. We also summarize a generic GNN-SD framework that could be exploited to induce other distillation strategies. Experiments further prove the effectiveness and generalization of our approach, as it brings: 1) state-of-the-art GNN distillation performance with less training cost, 2) consistent and considerable performance enhancement for various popular backbones.

REGION COLORING, EDGE COLORING, AND SCAN-CONVERSION OF MAPS

1994 ◽  
Vol 04 (04) ◽  
pp. 423-455
Author(s):  
KEITH D. McCROAN ◽  
R. C. LACHER
Keyword(s):  
Consistency Condition ◽  
Edge Coloring ◽  
High Complexity ◽  
Arbitrary Region ◽  
Embedded Graph ◽  
Original Region ◽  
Scan Conversion ◽  
Proof Of Correctness ◽  
Self Consistency ◽  
Very High

A theory is introduced relating extrinsic colorings of complementary regions of an embedded graph to certain intrinsic colorings of the edges of the graph, called color cycles, that satisfy a certain self-consistency condition. A region coloring is lifted to an edge coloring satisfying the cycle condition, and a dual construction builds a region coloring from any color cycle and any embedding of the graph. Both constructs are canonical, and the constructions are information-conservative in the sense that lifting an arbitrary region coloring to a color cycle and then reconstructing a region coloring from the cycle, using the same embedding, results in the original region coloring. The theory is motivated by, and provides the proof of correctness of, new scan-conversion algorithms that are useful in settings where region boundaries have very high complexity. These algorithms have been implemented and provide useful display functionality previously unavailable on certain rastor devices.

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