Geometric graph theory in the context of Geometric

Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles, spheres, polygons, and so forth. The subject focuses on the combinatorial properties of these objects, such as how they intersect one another, or how they may be arranged to cover a larger object.

Discrete geometry has a large overlap with convex geometry and computational geometry, and is closely related to subjects such as finite geometry, combinatorial optimization, digital geometry, discrete differential geometry, geometric graph theory, toric geometry, and combinatorial topology.

Graph drawing is an area of mathematics and computer science combining methods from geometric graph theory and information visualization to derive two-dimensional (or, sometimes, three-dimensional) depictions of graphs arising from applications such as social network analysis, cartography, linguistics, and bioinformatics.

A drawing of a graph or network diagram is a pictorial representation of the vertices and edges of a graph. This drawing should not be confused with the graph itself: very different layouts can correspond to the same graph. In the abstract, all that matters is which pairs of vertices are connected by edges. In the concrete, however, the arrangement of these vertices and edges within a drawing affects its understandability, usability, fabrication cost, and aesthetics. The problem gets worse if the graph changes over time by adding and deleting edges (dynamic graph drawing) and the goal is to preserve the user's mental map.

A spatial network (sometimes also geometric graph) is a graph in which the vertices or edges are spatial elements associated with geometric objects, i.e., the nodes are located in a space equipped with a certain metric. The simplest mathematical realization of spatial network is a lattice or a random geometric graph (see figure in the right), where nodes are distributed uniformly at random over a two-dimensional plane; a pair of nodes are connected if the Euclidean distance is smaller than a given neighborhood radius. Transportation and mobility networks, Internet, mobile phone networks, power grids, social and contact networks and biological neural networks are all examples where the underlying space is relevant and where the graph's topology alone does not contain all the information. Characterizing and understanding the structure, resilience and the evolution of spatial networks is crucial for many different fields ranging from urbanism to epidemiology.

In geometric graph theory, a unit disk graph is the intersection graph of a family of unit disks in the Euclidean plane. That is, it is a graph with one vertex for each disk in the family, and with an edge between two vertices whenever the corresponding vertices lie within a unit distance of each other.

They are commonly formed from a Poisson point process, making them a simple example of a random structure.

Aubrey David Nicholas Jasper de Grey (/dəˈɡreɪ/; born 20 April 1963) is an English biomedical gerontologist. He is the author of The Mitochondrial Free Radical Theory of Aging (1999) and co-author of Ending Aging (2007). De Grey is known for his view that medical technology may enable human beings alive today not to die from age-related causes. As an amateur mathematician, he has contributed to the study of the Hadwiger–Nelson problem in geometric graph theory, making the first progress on the problem in over 60 years.

De Grey is an international adjunct professor of the Moscow Institute of Physics and Technology. In August 2021, he was removed as the Chief Science Officer of the SENS Research Foundation after he had allegedly attempted to interfere in a probe investigating sexual harassment allegations against him. In September 2021, an independent investigation concluded that he had made offensive remarks to two women.

In geometric graph theory, a branch of mathematics, a polyhedral graph is the undirected graph formed from the vertices and edges of a convex polyhedron. Alternatively, in purely graph-theoretic terms, the polyhedral graphs are the 3-vertex-connected, planar graphs. The analogue concept for polytopes of general dimension are the polytopal graphs.