# Graph

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

# 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 discrete 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 graph theory. The word "graph" was first used in this sense by James 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 http://mathworld.wolfram.com/Graph.html

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 simple graphs. If multiple edges are allowed between vertices, the graph is known as a multigraph. Vertices are usually not allowed to be self-connected, but this restriction is sometimes relaxed to allow such "graph loops." A graph that may containmultiple edges and graph loops is called a pseudograph.

 GraphsLabeled

The edges, vertices, or both of a graph may be assigned specific values, labels, or colors, in which case the graph is called a labeled graph. A vertex 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 edge 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 graph 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 unlabeled graph.

 GraphsDirected

The edges of graphs may also be imbued with directedness. A normal graph in which edges are undirected is said to be undirected. 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 directed. A directed 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 oriented 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 network.

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