Intersection-Link Representations of Graphs

Pacific Journal of Mathematics, 1970

Lecture Notes in Computer Science, 1996

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Indonesian Journal of Combinatorics, 2021

Let G be a finite group and let N be a fixed normal subgroup of G. In this paper, a new kind of graph on G, namely the intersection graph is defined and studied. We use Γ in G (N) to denote this graph, with its vertices are all normal subgroups of G and two distinct vertices are adjacent if their intersection in N. We show some properties of this graph. For instance, the intersection graph is a simple connected with diameter at most two. Furthermore we give the graph structure of Γ in G ({e}) for some finite groups such as the symmetric, dihedral, special linear group, quaternion and cyclic groups.

1995

LINK is a set of C++ class libraries that supports applications in discrete mathematics. The libraries include a commandline interpreter and a graphical user interface that allow access to basic data structures such as Sets and Lists, and a graph hierarchy that includes undirected, directed, and \mixed" hypergraphs and graphs. A \mixed" graph may contain both directed and undirected edges. Many standard data structures including arrays, lists, heaps and binary search trees are within a Container hierarchy. Sets and Sequences are supported within a Collection hierarchy. The data structure hierarchies enable the user to experiment with competing data structure implementations, and with more complex and sophisticated data structures. If an algorithm has several possible choices of a data structure to be used, a single object can be created that is templated with the particular data structure desired. LINK also contains a set of graph generators, layout algorithms for hypergraphs and binary graphs, and numerous graph algorithms.

The inquiry briefly investigates similarities in patterns across a wide specter of organizational complexity – from simple cracks in dried mud to phenomena as complex as human habitats. All these patterns fall in the group of planar networks and can be represented as sets of nodes (vertices) and edges (links).

graph, 2018

In this first part of the book we develop some of the basic ideas behind graph theory-the study of network structure. This approach allows us to formulate basic network properties in a unifying language. The central definitions discussed here are simple enough that we can describe them relatively quickly at the outset; after this, we consider some fundamental applications of the definitions.

This paper presents a new framework for multivariate data analysis, based on graph theory, using intersection graphs . We have named this approach DAIG Data Analysis with Intersection Graphs. This new framework represents data vectors as paths on a graph, which has a number of advantages over the classical table representation of data. To do so, each node represents an atom of information, i.e. a pair of a variable and a value, associated with the set of observations for which that pair occurs. An edge exists between a pair of nodes whenever the intersection of their respective sets is not empty. We show that this representation of data as an intersection graph allows an easy and intuitive geometric interpretation of data observations, groups of observations, and results of multivariate data analysis techniques such as biplots, principal components, cluster analysis, or multidimensional scaling. These will appear as paths on the graph, relating variables, values and observations. This approach allows for a compact and memory efficient representation of data that contains many missing values or multi-valued attributes. The basic principles and advantages of this approach are presented with an example of its application to a simple toy problem. The main features of this methodology are illustrated with the aid software specifically developed for this purpose.

Intersection graphs are very important in both theoretical as well as application point of view. Depending on the geometrical representation, different type of intersection graphs are defined. Among them interval, circular-arc, permutation, trapezoid, chordal, disk, circle graphs are more important. In this article, a brief introduction of each of these intersection graphs is given. Some basic properties and algorithmic status of few problems on these graphs are cited. This article will help to the beginners to start work in this direction. Since the article contains a lot of information in a compact form it is also useful for the expert researchers too.

Annali di Matematica Pura ed Applicata (1923 -), 2016

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 welldeveloped 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.