This paper presents some adaptive restart randomized greedy heuristics formaximum clique. The algorithms are based on improvements and variations of previously-studied algorithms by the authors. Three kinds of adaptation are studied: adaptation of the initial state (AI) given to the greedy heuristic, adaptation of vertex weights (AW) on the graph, and no adaptation (NA). Two kinds of initializa...

Biological and cellular systems are often modeled as graphs in which vertices represent objects of interest (genes, proteins, drugs) and edges represent relational ties among these objects (binds-to, interacts-with, regulates). This approach has been highly successful owing to the theory, methodology and software that support analysis and learning on graphs. Graphs, however, often suffer from i...

The vertex connectivity of a graph is the smallest number of vertices whose deletion separates the graph or makes it trivial. This work is devoted to the problem of vertex connectivity test of graphs in a distributed environment based on a general and a constructive approach. The contribution of this paper is threefold. First, using a pre-constructed spanning tree of the considered graph, we pr...

We study properties of a C2-cofinite vertex operator algebra V = ⊕ ∞ i=0Vi of CFT type. If it is also rational (i.e. all modules are completely reducible) and V ′ ∼= V , then the rigidity of the tensor category of modules has been proved by Huang [11], where V ′ denotes the restricted dual of V . However, when we treat irrational C2cofinite VOAs, the rigidity is too strong, because it is almost...

We review 28 uniform partitions of 3-space in order to find out which of them have graphs (skeletons) embeddable isometrically (or with scale 2) into some cubic lattice Zn . We also consider some relatives of those 28 partitions, including Archimedean 4-polytopes of Conway–Guy, non-compact uniform partitions, Kelvin partitions and those with a unique vertex figure (i.e., Delaunay star). Among t...

This paper elaborates on the bulk/boundary relation between negative cosmological constant 3D gravity and Liouville field theory (LFT). We develop an interpretation of LFT non-normalizable states in terms of particles moving in the bulk. This interpretation is suggested by the fact that “heavy” vertex operators of LFT create conical singularities and thus should correspond to point particles mo...

We propose two new lower bounds on graph bandwidth and cyclic bandwidth based on semidefinite programming (SDP) relaxations of the quadratic assignment problem. We compare the new bounds with two other SDP bounds in [A. Blum, G. Konjevod, R. Ravi, and S. Vempala, Semi-definite relaxations for minimum bandwidth and other vertex-ordering problems, Theoretical Computer Science, 235(1):25-42, 2000]...

We consider the triangle-free process: given an integer n, start by taking a uniformly random ordering of the edges of the complete n-vertex graph Kn. Then, traverse the ordered edges and add each traversed edge to an (initially empty) evolving graph unless its addition creates a triangle. We study the evolving graph at around the time where n edges have been traversed for any fixed ε ∈ (0, 10)...