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The adjacency matrix of a graph as a data table: a geometric perspective. (English) Zbl 1366.05029
Summary: In this paper we continue a research project concerning the study of a graph from the perspective of granular computation. To be more specific, we interpret the adjacency matrix of any simple undirected graph $$G$$ in terms of data information table, which is one of the most studied structures in database theory. Granular computing (abbreviated GrC) is a well-developed research field in applied and theoretical information sciences; nevertheless, in this paper we address our efforts toward a purely mathematical development of the link between GrC and graph theory. From this perspective, the well-studied notion of indiscernibility relation in GrC becomes a symmetry relation with respect to a given vertex subset in graph theory; therefore, the investigation of this symmetry relation turns out to be the main object of study in this paper. In detail, we study a simple undirected graph $$G$$ by assuming a generic vertex subset $$W$$ as reference system with respect to which examine the symmetry of all vertex subsets of $$G$$. The change of perspective from $$G$$ without reference system to the pair $$(G, W)$$ is similar to what occurs in the transition from an affine space to a vector space. We interpret the symmetry blocks in the reference system $$(G, W)$$ as particular equivalence classes of vertices in $$G$$, and we study the geometric properties of all reference systems $$(G, W)$$, when $$W$$ runs over all vertex subsets of $$G$$. We also introduce three hypergraph models and a vertex set partition lattice associated to $$G$$, by taking as general models of reference several classical notions of GrC. For all these constructions, we provide a geometric characterization and we determine their structure for basic graph families. Finally, we apply a wide part of our work to study the important case of the Petersen graph.

##### MSC:
 05C10 Planar graphs; geometric and topological aspects of graph theory 05C65 Hypergraphs 05A18 Partitions of sets 05C75 Structural characterization of families of graphs 06A15 Galois correspondences, closure operators (in relation to ordered sets)
##### Keywords:
set partitions; lattices; hypergraphs; data tables
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##### References:
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