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In mathematics and computer science, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices (also called nodes or points) which are connected by edges (also called arcs, links or lines).
[1] [2] 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.
In discrete mathematics, particularly in graph theory, a graph is a structure consisting of a set of objects where some pairs of the objects are in some sense "related". The objects are represented by abstractions called vertices (also called nodes or points) and each of the related pairs of vertices is called an edge (also called link or line ...
The edge bipartization problem is the algorithmic problem of deleting as few edges as possible to make a graph bipartite and is also an important problem in graph modification algorithmics. This problem is also fixed-parameter tractable , and can be solved in time O ( 2 k m 2 ) {\textstyle O\left(2^{k}m^{2}\right)} , [ 33 ] where k is the ...
Any graph (which need not be simple; loops and multiple edges are allowed) is a uniform incidence structure with two points per line. For these examples, the vertices of the graph form the point set, the edges of the graph form the line set, and incidence means that a vertex is an endpoint of an edge.
Mathematical visualization is used throughout mathematics, particularly in the fields of geometry and analysis. Notable examples include plane curves , space curves , polyhedra , ordinary differential equations , partial differential equations (particularly numerical solutions, as in fluid dynamics or minimal surfaces such as soap films ...
In the case of a line arrangement, each coordinate of the labeling assigns 0 to nodes on one side of one of the lines and 1 to nodes on the other side. [26] Dual graphs of simplicial arrangements have been used to construct infinite families of 3-regular partial cubes, isomorphic to the graphs of simple zonohedra .
The graph K 3,3, for example, has 6 vertices, 9 edges, and no cycles of length 3. Therefore, by Theorem 2, it cannot be planar. Therefore, by Theorem 2, it cannot be planar. These theorems provide necessary conditions for planarity that are not sufficient conditions, and therefore can only be used to prove a graph is not planar, not that it is ...
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