Convex Graph Invariant Relaxations For Graph Edit Distance

The edit distance between two graphs is a widely used measure of similarity that evaluates the smallest number of vertex and edge deletions/insertions required to transform one graph to another. It is NP-hard to compute in general, and a large number of heuristics have been proposed for approximating this quantity. With few exceptions, these methods generally provide upper bounds on the edit distance between two graphs. In this paper, we propose a new family of computationally tractable convex relaxations for obtaining lower bounds on graph edit distance. These relaxations can be tailored to the structural properties of the particular graphs via convex graph invariants. Specific examples that we highlight in this paper include constraints on the graph spectrum as well as (tractable approximations of) the stability number and the maximum-cut values of graphs. We prove under suitable conditions that our relaxations are tight (i.e., exactly compute the graph edit distance) when one of the graphs consists of few eigenvalues. We also validate the utility of our framework on synthetic problems as well as real applications involving molecular structure comparison problems in chemistry.Comment: 27 pages, 7 figure

Denjoy constructions for fibred homeomorphisms of the torus

We construct different types of quasiperiodically forced circle homeomorphisms with transitive but non-minimal dynamics. Concerning the recent Poincar\'e-like classification for this class of maps of Jaeger-Stark, we demonstrate that transitive but non-minimal behaviour can occur in each of the different cases. This closes one of the last gaps in the topological classification. Actually, we are able to get some transitive quasiperiodically forced circle homeomorphisms with rather complicated minimal sets. For example, we show that, in some of the examples we construct, the unique minimal set is a Cantor set and its intersection with each vertical fibre is uncountable and nowhere dense (but may contain isolated points). We also prove that minimal sets of the later kind cannot occur when the dynamics are given by the projective action of a quasiperiodic SL(2,R)-cocycle. More precisely, we show that, for a quasiperiodic SL(2,R)-cocycle, any minimal strict subset of the torus either is a union of finitely many continuous curves, or contains at most two points on generic fibres

Distance-regular graphs

This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page