Knowledge representation through graphs

Due to the increasing amount of data, knowledge aggregation, representation and reasoning are highly important for companies. In this paper, knowledge aggregation is presented as the first step. In the sequel, successful knowledge representation, for instance through graphs, enables knowledge-based reasoning. There exist various forms of knowledge representation through graphs; some of which allow to handle uncertainty and imprecision by invoking the technology of fuzzy sets. The paper provides an overview of different types of graphs stressing their relationships and their essential features.

Bipartite graphs and quasipositive surfaces

We present a simple combinatorial model for quasipositive surfaces and positive braids, based on embedded bipartite graphs. As a first application, we extend the well-known duality on standard diagrams of torus links to twisted torus links. We then introduce a combinatorial notion of adjacency for bipartite graph links and discuss its potential relation with the adjacency problem for plane curve singularities.

Non-self-adjoint graphs

On finite metric graphs we consider Laplace operators, subject to various classes of non-self-adjoint boundary conditions imposed at graph vertices. We investigate spectral properties, existence of a Riesz basis of projectors and similarity transforms to self-adjoint Laplacians. Among other things, we describe a simple way to relate the similarity transforms between Laplacians on certain graphs with elementary similarity transforms between matrices defining the boundary conditions.