# Graph

```A graph is a mathematical representational tool. Read here about the use of mathematical graphs in tensegrity research.
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# Overview of Graphs

In the mathematical representation of tensegrity structures, a graph is an abstract representation of the set of objects that compose the tensegrity structure, where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called //vertices//, and the links that connect some pairs of vertices are called //edges//. Typically, a graph is depicted in diagrammatic form as a set of dots for the vertices, joined by lines or curves for the edges. Graphs are one of the objects of study in [mathematics].

Graph edges may be directed (asymmetric) or undirected (symmetric). Vertices are also called //nodes// or //points//, and edges are also called //lines//. Graphs are the basic subject studied by [theory]. The word "graph" was first used in this sense by [Joseph Sylvester] in 1878.

Below is an introduction to graph notation and how it helps explicate tensegrity structures.

## Overview of Graphs from Mathworld

The text below is an extract from []

Graphs come in a wide variety of different sorts. The most common type is graphs in which at most one edge (i.e., either one edge or no edges) may connect any two vertices. Such graphs are called [graphs]. If multiple edges are allowed between vertices, the graph is known as a []. Vertices are usually not allowed to be self-connected, but this restriction is sometimes relaxed to allow such "loops]." A graph that may containedges] and http://mathworld.wolfram.com/GraphLoop.html%7Cgraph loops]] is called a [[http://mathworld.wolfram.com/Pseudograph.html%7Cpseudograph.

The edges, vertices, or both of a graph may be assigned specific values, labels, or colors, in which case the graph is called a [graph]. A [coloring] is an assignment of labels or colors to each vertex of a graph such that no edge connects two identically colored vertices. Similarly, an [coloring] is an assignment of labels or colors to each edge of a graph such that adjacent edges (or the edges bounding different regions) must receive different colors. The assignment of labels or colors to the edges or vertices of a graph based on a set of specified criteria is known as [coloring]. If labels or colors are not permitted so that edges and vertices do not carry any additional properties beyond their intrinsic connectivities, a graph is called an [graph].

The edges of graphs may also be imbued with directedness. A normal graph in which edges are undirected is said to be []. Otherwise, if arrows may be placed on one or both endpoints of the edges of a graph to indicate directedness, the graph is said to be []. A [graph] in which each edge is given a unique direction (i.e., edges may not be bidirected and point on both directions as once) is called an [graph]. A graph or directed graph together with a function which assigns a positive real number to each edge (i.e., an oriented edge-labeled graph) is known as a [].