Graph theory

A graph with three vertices and three edges.

In one restricted but very common sense of the term,[1][2] a graph is an ordered pair ${\displaystyle G=(V,E)}$ comprising:

• ${\displaystyle V}$, a set of vertices (also called nodes or points);
• ${\displaystyle E\subseteq \{\{x,y\}\mid x,y\in V\;{\textrm {and}}\;x\neq y\}}$, a set of edges (also called links or lines), which are unordered pairs of vertices (that is, an edge is associated with two distinct vertices).

To avoid ambiguity, this type of object may be called precisely an undirected simple graph.

In the edge ${\displaystyle \{x,y\}}$, the vertices ${\displaystyle x}$ and ${\displaystyle y}$ are called the endpoints of the edge. The edge is said to join ${\displaystyle x}$ and ${\displaystyle y}$ and to be incident on ${\displaystyle x}$ and on ${\displaystyle y}$. A vertex may exist in a graph and not belong to an edge. Multiple edges, not allowed under the definition above, are two or more edges that join the same two vertices.

In one more general sense of the term allowing multiple edges,[3][4] a graph is an ordered triple ${\displaystyle G=(V,E,\phi )}$ comprising:

• ${\displaystyle V}$, a set of vertices (also called nodes or points);
• ${\displaystyle E}$, a set of edges (also called links or lines);
• ${\displaystyle \phi :E\to \{\{x,y\}\mid x,y\in V\;{\textrm {and}}\;x\neq y\}}$, an incidence function mapping every edge to an unordered pair of vertices (that is, an edge is associated with two distinct vertices).

To avoid ambiguity, this type of object may be called precisely an undirected multigraph.

A loop is an edge that joins a vertex to itself. Graphs as defined in the two definitions above cannot have loops, because a loop joining a vertex ${\displaystyle x}$ to itself is the edge (for an undirected simple graph) or is incident on (for an undirected multigraph) ${\displaystyle \{x,x\}=\{x\}}$ which is not in ${\displaystyle \{\{x,y\}\mid x,y\in V\;{\textrm {and}}\;x\neq y\}}$. So to allow loops the definitions must be expanded. For undirected simple graphs, the definition of ${\displaystyle E}$ should be modified to ${\displaystyle E\subseteq \{\{x,y\}\mid x,y\in V\}}$. For undirected multigraphs, the definition of ${\displaystyle \phi }$ should be modified to ${\displaystyle \phi :E\to \{\{x,y\}\mid x,y\in V\}}$. To avoid ambiguity, these types of objects may be called undirected simple graph permitting loops and undirected multigraph permitting loops (sometimes also undirected pseudograph), respectively.

${\displaystyle V}$ and ${\displaystyle E}$ are usually taken to be finite, and many of the well-known results are not true (or are rather different) for infinite graphs because many of the arguments fail in the infinite case. Moreover, ${\displaystyle V}$ is often assumed to be non-empty, but ${\displaystyle E}$ is allowed to be the empty set. The order of a graph is ${\displaystyle |V|}$, its number of vertices. The size of a graph is ${\displaystyle |E|}$, its number of edges. The degree or valency of a vertex is the number of edges that are incident to it, where a loop is counted twice. The degree of a graph is the maximum of the degrees of its vertices.

In an undirected simple graph of order n, the maximum degree of each vertex is n − 1 and the maximum size of the graph is n(n − 1)/2.

The edges of an undirected simple graph permitting loops ${\displaystyle G}$ induce a symmetric homogeneous relation ${\displaystyle \sim }$ on the vertices of ${\displaystyle G}$ that is called the adjacency relation of ${\displaystyle G}$. Specifically, for each edge ${\displaystyle (x,y)}$, its endpoints ${\displaystyle x}$ and ${\displaystyle y}$ are said to be adjacent to one another, which is denoted ${\displaystyle x\sim y}$.

A directed graph with three vertices and four directed edges (the double arrow represents an edge in each direction).

A directed graph or digraph is a graph in which edges have orientations.

In one restricted but very common sense of the term,[5] a directed graph is an ordered pair ${\displaystyle G=(V,E)}$ comprising:

• ${\displaystyle V}$, a set of vertices (also called nodes or points);
• ${\displaystyle E\subseteq \left\{(x,y)\mid (x,y)\in V^{2}\;{\textrm {and}}\;x\neq y\right\}}$, a set of edges (also called directed edges, directed links, directed lines, arrows or arcs) which are ordered pairs of vertices (that is, an edge is associated with two distinct vertices).

To avoid ambiguity, this type of object may be called precisely a directed simple graph.

In the edge ${\displaystyle (x,y)}$ directed from ${\displaystyle x}$ to ${\displaystyle y}$, the vertices ${\displaystyle x}$ and ${\displaystyle y}$ are called the endpoints of the edge, ${\displaystyle x}$ the tail of the edge and ${\displaystyle y}$ the head of the edge. The edge is said to join ${\displaystyle x}$ and ${\displaystyle y}$ and to be incident on ${\displaystyle x}$ and on ${\displaystyle y}$. A vertex may exist in a graph and not belong to an edge. The edge ${\displaystyle (y,x)}$ is called the inverted edge of ${\displaystyle (x,y)}$. Multiple edges, not allowed under the definition above, are two or more edges with both the same tail and the same head.

In one more general sense of the term allowing multiple edges,[5] a directed graph is an ordered triple ${\displaystyle G=(V,E,\phi )}$ comprising:

• ${\displaystyle V}$, a set of vertices (also called nodes or points);
• ${\displaystyle E}$, a set of edges (also called directed edges, directed links, directed lines, arrows or arcs);
• ${\displaystyle \phi :E\to \left\{(x,y)\mid (x,y)\in V^{2}\;{\textrm {and}}\;x\neq y\right\}}$, an incidence function mapping every edge to an ordered pair of vertices (that is, an edge is associated with two distinct vertices).

To avoid ambiguity, this type of object may be called precisely a directed multigraph.

A loop is an edge that joins a vertex to itself. Directed graphs as defined in the two definitions above cannot have loops, because a loop joining a vertex ${\displaystyle x}$ to itself is the edge (for a directed simple graph) or is incident on (for a directed multigraph) ${\displaystyle (x,x)}$ which is not in ${\displaystyle \left\{(x,y)\mid (x,y)\in V^{2}\;{\textrm {and}}\;x\neq y\right\}}$. So to allow loops the definitions must be expanded. For directed simple graphs, the definition of ${\displaystyle E}$ should be modified to ${\displaystyle E\subseteq \left\{(x,y)\mid (x,y)\in V^{2}\right\}}$. For directed multigraphs, the definition of ${\displaystyle \phi }$ should be modified to ${\displaystyle \phi :E\to \left\{(x,y)\mid (x,y)\in V^{2}\right\}}$. To avoid ambiguity, these types of objects may be called precisely a directed simple graph permitting loops and a directed multigraph permitting loops (or a quiver) respectively.

The edges of a directed simple graph permitting loops ${\displaystyle G}$ is a homogeneous relation ~ on the vertices of ${\displaystyle G}$ that is called the adjacency relation of ${\displaystyle G}$. Specifically, for each edge ${\displaystyle (x,y)}$, its endpoints ${\displaystyle x}$ and ${\displaystyle y}$ are said to be adjacent to one another, which is denoted ${\displaystyle x}$ ~ ${\displaystyle y}$.

Within computer science, cybernetics uses graphs to represent networks of communication, data organization, computational devices, the flow of computation, etc. For instance, the link structure of a website can be represented by a directed graph, in which the vertices represent web pages and directed edges represent links from one page to another. A similar approach can be taken to problems in social media,[10] travel, biology, computer chip design, mapping the progression of neuro-degenerative diseases,[11][12] and many other fields. The development of algorithms to handle graphs is therefore of major interest in computer science. The transformation of graphs is often formalized and represented by graph rewrite systems. Complementary to graph transformation systems focusing on rule-based in-memory manipulation of graphs are graph databases geared towards transaction-safe, persistent storing and querying of graph-structured data.

Graph-theoretic methods, in various forms, have proven particularly useful in linguistics, since natural language often lends itself well to discrete structure. Traditionally, syntax and compositional semantics follow tree-based structures, whose expressive power lies in the principle of compositionality, modeled in a hierarchical graph. More contemporary approaches such as head-driven phrase structure grammar model the syntax of natural language using typed feature structures, which are directed acyclic graphs. Within lexical semantics, especially as applied to computers, modeling word meaning is easier when a given word is understood in terms of related words; semantic networks are therefore important in computational linguistics. Still, other methods in phonology (e.g. optimality theory, which uses lattice graphs) and morphology (e.g. finite-state morphology, using finite-state transducers) are common in the analysis of language as a graph. Indeed, the usefulness of this area of mathematics to linguistics has borne organizations such as TextGraphs, as well as various 'Net' projects, such as WordNet, VerbNet, and others.

Graph theory is also used to study molecules in chemistry and physics. In condensed matter physics, the three-dimensional structure of complicated simulated atomic structures can be studied quantitatively by gathering statistics on graph-theoretic properties related to the topology of the atoms. Also, "the Feynman graphs and rules of calculation summarize quantum field theory in a form in close contact with the experimental numbers one wants to understand."[13] In chemistry a graph makes a natural model for a molecule, where vertices represent atoms and edges bonds. This approach is especially used in computer processing of molecular structures, ranging from chemical editors to database searching. In statistical physics, graphs can represent local connections between interacting parts of a system, as well as the dynamics of a physical process on such systems. Similarly, in computational neuroscience graphs can be used to represent functional connections between brain areas that interact to give rise to various cognitive processes, where the vertices represent different areas of the brain and the edges represent the connections between those areas. Graph theory plays an important role in electrical modeling of electrical networks, here, weights are associated with resistance of the wire segments to obtain electrical properties of network structures.[14] Graphs are also used to represent the micro-scale channels of porous media, in which the vertices represent the pores and the edges represent the smaller channels connecting the pores. Chemical graph theory uses the molecular graph as a means to model molecules. Graphs and networks are excellent models to study and understand phase transitions and critical phenomena. Removal of nodes or edges leads to a critical transition where the network breaks into small clusters which is studied as a phase transition. This breakdown is studied via percolation theory.[15]

Graph theory in sociology: Moreno Sociogram (1953).[16]

Graph theory is also widely used in sociology as a way, for example, to measure actors' prestige or to explore rumor spreading, notably through the use of social network analysis software. Under the umbrella of social networks are many different types of graphs.[17] Acquaintanceship and friendship graphs describe whether people know each other. Influence graphs model whether certain people can influence the behavior of others. Finally, collaboration graphs model whether two people work together in a particular way, such as acting in a movie together.

Likewise, graph theory is useful in biology and conservation efforts where a vertex can represent regions where certain species exist (or inhabit) and the edges represent migration paths or movement between the regions. This information is important when looking at breeding patterns or tracking the spread of disease, parasites or how changes to the movement can affect other species.

Graphs are also commonly used in molecular biology and genomics to model and analyse datasets with complex relationships. For example, graph-based methods are often used to 'cluster' cells together into cell-types in single-cell transcriptome analysis. Another use is to model genes or proteins in a pathway and study the relationships between them, such as metabolic pathways and gene regulatory networks.[18] Evolutionary trees, ecological networks, and hierarchical clustering of gene expression patterns are also represented as graph structures.

Graph theory is also used in connectomics;[19] nervous systems can be seen as a graph, where the nodes are neurons and the edges are the connections between them.

In mathematics, graphs are useful in geometry and certain parts of topology such as knot theory. Algebraic graph theory has close links with group theory. Algebraic graph theory has been applied to many areas including dynamic systems and complexity.

A graph structure can be extended by assigning a weight to each edge of the graph. Graphs with weights, or weighted graphs, are used to represent structures in which pairwise connections have some numerical values. For example, if a graph represents a road network, the weights could represent the length of each road. There may be several weights associated with each edge, including distance (as in the previous example), travel time, or monetary cost. Such weighted graphs are commonly used to program GPS's, and travel-planning search engines that compare flight times and costs.

Graphs are usually represented visually by drawing a point or circle for every vertex, and drawing a line between two vertices if they are connected by an edge. If the graph is directed, the direction is indicated by drawing an arrow. If the graph is weighted, the weight is added on the arrow.

A graph drawing should not be confused with the graph itself (the abstract, non-visual structure) as there are several ways to structure the graph drawing. All that matters is which vertices are connected to which others by how many edges and not the exact layout. In practice, it is often difficult to decide if two drawings represent the same graph. Depending on the problem domain some layouts may be better suited and easier to understand than others.

The pioneering work of W. T. Tutte was very influential on the subject of graph drawing. Among other achievements, he introduced the use of linear algebraic methods to obtain graph drawings.

Graph drawing also can be said to encompass problems that deal with the crossing number and its various generalizations. The crossing number of a graph is the minimum number of intersections between edges that a drawing of the graph in the plane must contain. For a planar graph, the crossing number is zero by definition. Drawings on surfaces other than the plane are also studied.

There are other techniques to visualize a graph away from vertices and edges, including circle packings, intersection graph, and other visualizations of the adjacency matrix.

The tabular representation lends itself well to computational applications. There are different ways to store graphs in a computer system. The data structure used depends on both the graph structure and the algorithm used for manipulating the graph. Theoretically one can distinguish between list and matrix structures but in concrete applications the best structure is often a combination of both. List structures are often preferred for sparse graphs as they have smaller memory requirements. Matrix structures on the other hand provide faster access for some applications but can consume huge amounts of memory. Implementations of sparse matrix structures that are efficient on modern parallel computer architectures are an object of current investigation.[33]

List structures include the edge list, an array of pairs of vertices, and the adjacency list, which separately lists the neighbors of each vertex: Much like the edge list, each vertex has a list of which vertices it is adjacent to.

Matrix structures include the incidence matrix, a matrix of 0's and 1's whose rows represent vertices and whose columns represent edges, and the adjacency matrix, in which both the rows and columns are indexed by vertices. In both cases a 1 indicates two adjacent objects and a 0 indicates two non-adjacent objects. The degree matrix indicates the degree of vertices. The Laplacian matrix is a modified form of the adjacency matrix that incorporates information about the degrees of the vertices, and is useful in some calculations such as Kirchhoff's theorem on the number of spanning trees of a graph. The distance matrix, like the adjacency matrix, has both its rows and columns indexed by vertices, but rather than containing a 0 or a 1 in each cell it contains the length of a shortest path between two vertices.

There is a large literature on graphical enumeration: the problem of counting graphs meeting specified conditions. Some of this work is found in Harary and Palmer (1973).

A common problem, called the subgraph isomorphism problem, is finding a fixed graph as a subgraph in a given graph. One reason to be interested in such a question is that many graph properties are hereditary for subgraphs, which means that a graph has the property if and only if all subgraphs have it too. Unfortunately, finding maximal subgraphs of a certain kind is often an NP-complete problem. For example:

• Finding the largest complete subgraph is called the clique problem (NP-complete).

One special case of subgraph isomorphism is the graph isomorphism problem. It asks whether two graphs are isomorphic. It is not known whether this problem is NP-complete, nor whether it can be solved in polynomial time.

A similar problem is finding induced subgraphs in a given graph. Again, some important graph properties are hereditary with respect to induced subgraphs, which means that a graph has a property if and only if all induced subgraphs also have it. Finding maximal induced subgraphs of a certain kind is also often NP-complete. For example:

• Finding the largest edgeless induced subgraph or independent set is called the independent set problem (NP-complete).

Still another such problem, the minor containment problem, is to find a fixed graph as a minor of a given graph. A minor or subcontraction of a graph is any graph obtained by taking a subgraph and contracting some (or no) edges. Many graph properties are hereditary for minors, which means that a graph has a property if and only if all minors have it too. For example, Wagner's Theorem states:

• A graph is planar if it contains as a minor neither the complete bipartite graph K_3,3 (see the Three-cottage problem) nor the complete graph K₅.

A similar problem, the subdivision containment problem, is to find a fixed graph as a subdivision of a given graph. A subdivision or homeomorphism of a graph is any graph obtained by subdividing some (or no) edges. Subdivision containment is related to graph properties such as planarity. For example, Kuratowski's Theorem states:

• A graph is planar if it contains as a subdivision neither the complete bipartite graph K_3,3 nor the complete graph K₅.

Another problem in subdivision containment is the Kelmans–Seymour conjecture:

• Every 5-vertex-connected graph that is not planar contains a subdivision of the 5-vertex complete graph K₅.

Another class of problems has to do with the extent to which various species and generalizations of graphs are determined by their point-deleted subgraphs. For example:

• The reconstruction conjecture

Many problems and theorems in graph theory have to do with various ways of coloring graphs. Typically, one is interested in coloring a graph so that no two adjacent vertices have the same color, or with other similar restrictions. One may also consider coloring edges (possibly so that no two coincident edges are the same color), or other variations. Among the famous results and conjectures concerning graph coloring are the following:

• Four-color theorem
• Strong perfect graph theorem
• Erdős–Faber–Lovász conjecture
• Total coloring conjecture, also called Behzad's conjecture (unsolved)
• List coloring conjecture (unsolved)
• Hadwiger conjecture (graph theory) (unsolved)

Constraint modeling theories concern families of directed graphs related by a partial order. In these applications, graphs are ordered by specificity, meaning that more constrained graphs—which are more specific and thus contain a greater amount of information—are subsumed by those that are more general. Operations between graphs include evaluating the direction of a subsumption relationship between two graphs, if any, and computing graph unification. The unification of two argument graphs is defined as the most general graph (or the computation thereof) that is consistent with (i.e. contains all of the information in) the inputs, if such a graph exists; efficient unification algorithms are known.

For constraint frameworks which are strictly compositional, graph unification is the sufficient satisfiability and combination function. Well-known applications include automatic theorem proving and modeling the elaboration of linguistic structure.

• Hamiltonian path problem
• Minimum spanning tree
• Route inspection problem (also called the "Chinese postman problem")
• Seven bridges of Königsberg
• Shortest path problem
• Steiner tree
• Three-cottage problem
• Traveling salesman problem (NP-hard)

There are numerous problems arising especially from applications that have to do with various notions of flows in networks, for example:

• Max flow min cut theorem

• Museum guard problem

Covering problems in graphs may refer to various set cover problems on subsets of vertices/subgraphs.

• Dominating set problem is the special case of set cover problem where sets are the closed neighborhoods.
• Vertex cover problem is the special case of set cover problem where sets to cover are every edges.
• The original set cover problem, also called hitting set, can be described as a vertex cover in a hypergraph.

Decomposition, defined as partitioning the edge set of a graph (with as many vertices as necessary accompanying the edges of each part of the partition), has a wide variety of questions. Often, the problem is to decompose a graph into subgraphs isomorphic to a fixed graph; for instance, decomposing a complete graph into Hamiltonian cycles. Other problems specify a family of graphs into which a given graph should be decomposed, for instance, a family of cycles, or decomposing a complete graph K_n into n − 1 specified trees having, respectively, 1, 2, 3, ..., n − 1 edges.

Some specific decomposition problems that have been studied include:

• Arboricity, a decomposition into as few forests as possible
• Cycle double cover, a decomposition into a collection of cycles covering each edge exactly twice
• Edge coloring, a decomposition into as few matchings as possible
• Graph factorization, a decomposition of a regular graph into regular subgraphs of given degrees

Many problems involve characterizing the members of various classes of graphs. Some examples of such questions are below:

• Enumerating the members of a class
• Characterizing a class in terms of forbidden substructures
• Ascertaining relationships among classes (e.g. does one property of graphs imply another)
• Finding efficient algorithms to decide membership in a class
• Finding representations for members of a class

• Phase Transitions in Combinatorial Optimization Problems, Section 3: Introduction to Graphs (2006) by Hartmann and Weigt
• Digraphs: Theory Algorithms and Applications 2007 by Jorgen Bang-Jensen and Gregory Gutin
• Graph Theory, by Reinhard Diestel