Structures that may be represented when graphical record come omnipresent, & numerous problems of practical interest may be represented by graphical record. A hyperlink structure of Wikipedia could be represented by a directed graphical record: a vertices come a articles inside Wikipedia & there's the directed edge from either article The to article B in case & single in case The contains the return B. A development of algorithms to handle graphical record is so of major interest around computer science.

The graphical record structure may be extended by assigning the weight to both edge. Graphical record sustaining weights may be utilized to represent several different conception; e.g. in case a graphical record is a touring network, a weights can represent the length of every road¹. The second way to extend basic graphical record is by making a edges to the graphical record directing (A links to B, however B doesn't necessarily return The, every bit inside webpages), technically known as the directed graph or digraph. The digram by owning weighted edges is known as the network.

Networks keep close at hand numbers of utilizes in the practical side of graphical record theory, network analysis (for even example, to model & analyze traffic networks or to discover a shape of the internet -- understand Applications below). Yet, it should exist as noted that in network analysis, a definition of the term "network" might differ, & will typically refer to the elementary graphical record.

History
One of a number one resolutions inside graphical record theory appeared within Leonhard Euler's paper on Seven Bridges of Königsberg, published in 1736. These are likewise repute one of a 1st topologic effects within geometry; that is, it doesn't depend in any mensuration. This illustrates a deep connection between graphical record theory & topology.

Inside 1845 Gustav Kirchhoff published his Kirchhoff's circuit laws for calculating the voltage and current in electric circuits.

Around 1852 Francis Guthrie posed the four color problem which asks if these are conceivable to color, applying sole quaternary colors, any map of countries within such how else when to end 2 bordering countries from either getting a equivalent color. This condition, which was single solved the century within the future in 1976 by Kenneth Appel and Wolfgang Haken, can be considered a birth of graphical record theory. When trying to solve it mathematicians invented several fundamental graphical record theoretical terms & construct.

Definition

Drawing graphs

Graphical record come represented graphically by drawing the dot for each vertex, & drawing an arc between deuce vertices in case it is attached by an edge. Whenever a graphical record is directed a counsel is indicated by drawing an arrow.

a graphical record drawing should non become confused by owning a graphical record itself (a abstract, non-graphical structure) when there are many ways to structure the graphical record drawing. Completely that matters is which vertices come attached to which others by how else numerous edges & non a precise layout. Inside practise these are typically hard to decide whenever ii drawings represent a equivalent graphical record. Based on the condition domain occasionally layouts can be better suited & more leisurely to see than others.

Graphs as data structures

There are different ways to store graphical record around the adp system. A data structure used depends on each a graphical record structure & a algorithm used for manipulating the graphical record. Theoretically 1 potty distinguish between listings & matrix structures however around concrete applications the better structure is typically a combination of each. Listings structures come typically favorite for sparse graphs as they use little memory requirements. Matrix structures however then provide sooner access but might consume immense numbers of memory whenever a graphical record is very big.

List structures

Incidence list - The edges come represented by an array containing pairs (ordered in case directed) of vertices (that a edge connects) & finally weight & more information.

Adjacency list - Much such as the incidence listings, from each one node has a listing of which nodes these are adjacent to. This could occasionally symptom inside "overkill" inside an directionless graphical record when vertex Three can become in the names for node Deuce, so node Two must be in the listing for node Deuce-ace. Either a software engineer could pick out to utilise a unnecessary space anyway, or even he/she could see to listings a contiguousness another time. This representation is more easygoing to locate all the nodes which are then attached to one node, since which are actually explicitly used.

Matrix structures

Incidence matrix - The graphical record is represented by the matrix of E (edges) by V (vertices), in which [edge, vertex] contains a edge's information (simplest instance: One - attached, 0 - does'nt attached).

Adjacency matrix - there is aNorth North by North matrix, in which N is the total of vertices in the graphical record. Whenever there exists an edge from either a select few vertex x to a few vertex y, so a element $M\_$ is I, otherwise these are Nought. This makes it gentler to buy subgraphical record, & to reverse graphs whenever needful.

Admittance matrix or Kirchhoff matrix or Laplacian matrix - is defined as degree matrix minus adjacency matrix & thus contains contiguousness references and degree principles all about a vertices

Graph problems

Finding subgraphs

The most common condition, known as subgraph isomorphism problem, is finding subgraphs in a given graphical record. Numerous graph properties are hereditary, which means if a certain subgraphical record has the property thus does the altogether graph. E.g. the graphical record is non planar if it contains the complete bipartite graph $K\_$. Unluckily, selecting maximum subgraphs of the certain form is typically an NP-complete problem.

finding a big complete graph is called a clique problem (NP-complete) finding a big independent set is called a independent set problem (NP-complete)

Graph coloring
a four-color theorem a strong perfect graph theorem a Erdős-Faber-Lovász conjecture (unsolved) a total coloring conjecture (unsolved) a list coloring conjecture (unsolved)

Route problems
Seven Bridges of Königsberg Minimum spanning tree Steiner tree Shortest path problem Route inspection problem (also known as a "Chinese Postman Problem") Traveling salesman problem (NP-Complete)

Network flow
Max flow min cut theorem Reconstruction conjecture

Visibility graph problems
Museum guard problem

Covering Problems

Covering problems come specific cases of subgraph selecting problems, & tend to exist as closely related the clique problem or independent set problem.

Set cover problem Vertex cover problem

Important algorithms

Dijkstra's algorithm Kruskal's algorithm Nearest neighbour algorithm Prim's algorithm

Related areas of mathematics

Ramsey theory Combinatorics

Applications

Numbers of applications of graphical record theory survive in the form of network analysis. These split broadly into ii categories. First, analysis to determine structural properties of the network, like whether or even does'nt these are the scale-free network, or the small-world network. Second, analysis to call for the mensurable quantity inside the network, e.g., for a transportation network, the level of vehicular flow in any part of it.

Graphical record theory is too wont to learn molecules inside science. Within condensed matter physical science, a tierce miscreate structure of complicated simulated atomlike structures may be exposed quantitatively by gathering cost comparisons in graph-theoretic properties related to the topology of the atoms. E.g., Franzblau's shortest-path (SP) rings.

Subareas

Graphical record theory is diverse & contains numbers of identifiable subareas. A few of the babies come:

Algebraic graph theory Topological graph theory Geometric graph theory Extremal graph theory Metric graph theory

Prominent graph theorists

Paul Erdős Frank Harary Denes König W.T. Tutte [http://www1.cs.columbia.edu/~sanders/graphtheory/people/alphabetic.html Graph theory white pages] for further graphical record theoretician & their publications.

Notes