Finite Graphs and the Number of Sums and Products

An emerging technique in image segmentation, semi-supervised learning and general classification problems concerns the use of phase-separating flows defined on finite graphs. This technique was pioneered in Bertozzi and Flenner (2012, Multiscale Modeling and Simulation10(3), 1090–1118), which used the Allen–Cahn flow on a graph, and was then extended in Merkurjev et al. (2013, SIAM J. Imaging Sci.6(4), 1903–1930) using instead the Merriman–Bence–Osher (MBO) scheme on a graph. In previous work by the authors, Budd and Van Gennip (2020, SIAM J. Math. Anal.52(5), 4101–4139), we gave a theoretical justification for this use of the MBO scheme in place of Allen–Cahn flow, showing that the MBO scheme is a special case of a ‘semi-discrete’ numerical scheme for Allen–Cahn flow. In this paper, we extend this earlier work, showing that this link via the semi-discrete scheme is robust to passing to the mass-conserving case. Inspired by Rubinstein and Sternberg (1992, IMA J. Appl. Math.48, 249–264), we define a mass-conserving Allen–Cahn equation on a graph. Then, with the help of the tools of convex optimisation, we show that our earlier machinery can be applied to derive the mass-conserving MBO scheme on a graph as a special case of a semi-discrete scheme for mass-conserving Allen–Cahn. We give a theoretical analysis of this flow and scheme, proving various desired properties like existence and uniqueness of the flow and convergence of the scheme, and also show that the semi-discrete scheme yields a choice function for solutions to the mass-conserving MBO scheme.

Download Full-text

Arboricity and tree-packing in locally finite graphs

Journal of Combinatorial Theory Series B ◽

10.1016/j.jctb.2005.08.003 ◽

2006 ◽

Vol 96 (2) ◽

pp. 302-312 ◽

Cited By ~ 13

Author(s):

Maya Jakobine Stein

Keyword(s):

Locally Finite ◽

Finite Graphs ◽

Locally Finite Graphs ◽

Tree Packing

Download Full-text

Liftings in Finite Graphs and Linkages in Infinite Graphs with Prescribed Edge-Connectivity

Graphs and Combinatorics ◽

10.1007/s00373-016-1724-9 ◽

2016 ◽

Vol 32 (6) ◽

pp. 2575-2589

Author(s):

Seongmin Ok ◽

R. Bruce Richter ◽

Carsten Thomassen

Keyword(s):

Infinite Graphs ◽

Edge Connectivity ◽

Finite Graphs

Download Full-text

The edmonds—Gallai decomposition for matchings in locally finite graphs

COMBINATORICA ◽

10.1007/bf02579231 ◽

1982 ◽

Vol 2 (3) ◽

pp. 229-235 ◽

Cited By ~ 3

Author(s):

François Bry ◽

Michel Las Vergnas

Keyword(s):

Locally Finite ◽

Finite Graphs ◽

Locally Finite Graphs

Download Full-text

Approximating Cayley Diagrams Versus Cayley Graphs

Combinatorics Probability Computing ◽

10.1017/s0963548311000733 ◽

2012 ◽

Vol 21 (4) ◽

pp. 635-641

Author(s):

ÁDÁM TIMÁR

Keyword(s):

Cayley Graph ◽

Hamiltonian Cycle ◽

Cayley Graphs ◽

The Other ◽

Finite Graphs ◽

Similar Construction ◽

Graph Property

We construct a sequence of finite graphs that weakly converge to a Cayley graph, but there is no labelling of the edges that would converge to the corresponding Cayley diagram. A similar construction is used to give graph sequences that converge to the same limit, and such that a Hamiltonian cycle in one of them has a limit that is not approximable by any subgraph of the other. We give an example where this holds, but convergence is meant in a stronger sense. This is related to whether having a Hamiltonian cycle is a testable graph property.

Download Full-text