Miscellaneous properties of embeddings of line, total and middle graphs

Chartrand et al. (J. Combin. Theory Ser. B 10 (1971) 12–41) proved that the line graph of a graph G is outerplanar if and only if the total graph of G is planar. In this paper, we prove that these two conditions are equivalent to the middle graph of G been generalized outerplanar. Also, we show that a total graph is generalized outerplanar if and only if it is outerplanar. Later on, we characterize the graphs G such that Full-size image (<1 K) is planar, where Full-size image (<1 K) is a composition of the operations line, middle and total graphs. Also, we give an algorithm which decides whether or not Full-size image (<1 K) is planar in an Full-size image (<1 K) time, where n is the number of vertices of G. Finally, we give two characterizations of graphs so that their total and middle graphs admit an embedding in the projective plane. The first characterization shows the properties that a graph must verify in order to have a projective total and middle graph. The second one is in terms of forbidden subgraphs

The Spectrum of an Adelic Markov Operator

With the help of the representation of SL(2,Z) on the rank two free module over the integer adeles, we define the transition operator of a Markov chain. The real component of its spectrum exhibits a gap, whereas the non-real component forms a circle of radius 1/\sqrt{2}.Comment: 38 pages, 5 figure

A Comprehensive Study on Knowledge Graph Embedding over Relational Patterns Based on Rule Learning

Knowledge Graph Embedding (KGE) has proven to be an effective approach to solving the Knowledge Graph Completion (KGC) task. Relational patterns which refer to relations with specific semantics exhibiting graph patterns are an important factor in the performance of KGE models. Though KGE models' capabilities are analyzed over different relational patterns in theory and a rough connection between better relational patterns modeling and better performance of KGC has been built, a comprehensive quantitative analysis on KGE models over relational patterns remains absent so it is uncertain how the theoretical support of KGE to a relational pattern contributes to the performance of triples associated to such a relational pattern. To address this challenge, we evaluate the performance of 7 KGE models over 4 common relational patterns on 2 benchmarks, then conduct an analysis in theory, entity frequency, and part-to-whole three aspects and get some counterintuitive conclusions. Finally, we introduce a training-free method Score-based Patterns Adaptation (SPA) to enhance KGE models' performance over various relational patterns. This approach is simple yet effective and can be applied to KGE models without additional training. Our experimental results demonstrate that our method generally enhances performance over specific relational patterns. Our source code is available from GitHub at https://github.com/zjukg/Comprehensive-Study-over-Relational-Patterns.Comment: This paper is accepted by ISWC 202

A Diagrammatic Temperley-Lieb Categorification

The monoidal category of Soergel bimodules categorifies the Hecke algebra of a finite Weyl group. In the case of the symmetric group, morphisms in this category can be drawn as graphs in the plane. We define a quotient category, also given in terms of planar graphs, which categorifies the Temperley-Lieb algebra. Certain ideals appearing in this quotient are related both to the 1-skeleton of the Coxeter complex and to the topology of 2D cobordisms. We demonstrate how further subquotients of this category will categorify the cell modules of the Temperley-Lieb algebra.Comment: long awaited update to published versio