Topological graph theory Topological graph theory deals with the study of graphs as
topological spaces. The graph in a topology is a set of
simplexes that is called the
simplicial one-dimensional complex. This subarea studies the
embedding (or imbedding) of a graph in
surface and
linkless embedding,
graph minors,
crossing number, map coloring, and
voltage graph. The embedding of a graph in a surface is the representation of a graph in which the points are associated with the vertices and simple arcs with edges in a surface. The endpoints are associated with an edge, and the points with the end vertices. No arcs include points associated with other vertices, and two arcs never intersect at a point that is interior to either of the arcs. The graph embedding can be generalized into the linkless embedding, whereby no two cycles of the graph are linked in three-dimensional Euclidean space, and a
book, a collection of
half-planes all having the same line as their boundary. The graph is said to be minor if it can be formed from another graph by deleting vertices and edges, and by
edge-contraction. The earliest result of the graph minor theory is from
Wagner's theorem, stating that a finite graph is planar if and only if its minor includes neither the
complete graph on five vertices K_5 nor the
utility graph. A related result is the
Robertson–Seymour theorem, implying the existence of
forbidden minor for every property of graphs preserved by deletions and edge contractions. The crossing number tells the minimum number of crossing edges of a graph. This study originated from a Hungarian mathematician
Pál Turán who
asked for a factory plan that minimized the number of crossings between tracks connecting brick kilns to storage sites. This problem can be formalized as asking for the crossing number of a
complete bipartite graph. A
graph coloring is a methodical assignment of labelling the elements of a graph, which is traditionally called
colors. In coloring, no two adjacent elements have the same color. It requires the minimum number of colors, which is known as the chromatic number.
Four-color theorem stated that no more than four colors are required to color the regions of any map so that no two adjacent regions have the same color; that is, no two regions share a common boundary. This theorem is stronger than
five-color theorem. Relatedly,
Earth–Moon problem is known for the nowadays open problem on extension of the planar map coloring problem, solved by the four-color theorem. A voltage graph is a directed graph whose edges are labelled invertibly by elements of a
group. This graph concisely specifies the
derived graph. It is also a common way to form a
covering graph.
Algebraic graph theory is high-symmetrical, known for
vertex-transitive,
symmetric,
distance-transitive, and
distance-regular. Its
automorphism group has 120 elements and
symmetric group S_5.
Algebraic graph theory is the study of graph theory that involves major branches of
algebra. Major branches of algebra that are used are
linear algebra and
group theory. A study of graph theory using linear algebra is called
spectral graph theory. This study focuses on
adjacency matrix, a matrix that represents the graph, and its
spectrum, which focuses on the
characteristic polynomial,
eigenvalues, and eigenvectors of the given adjacency matrix. It also focuses on the
Laplacian matrix of a graph, which involves the
degree matrix (a
diagonal matrix that represents the degree of a vertex) and the adjacency matrix. Group theory, particularly
automorphism groups and
geometric group theory, focuses on various families of graphs based on symmetry in algebraic graph theory. Such a symmetry includes
symmetric graphs,
vertex-transitive graphs,
edge-transitive graphs,
distance-transitive graphs,
distance-regular graphs, and
strongly regular graphs.
Frucht's theorem states that every
finite group is the group of symmetries of a finite undirected graph, or more strongly, there exist infinitely many non-isomorphic simple connected graphs such that the automorphism group of each of them is isomorphic to a finite group. Algebraic graph theory also studies the algebraic
invariants,
chromatic polynomial,
Tutte polynomial of a graph, and
knot invariant. A graph invariant is a property of graphs that depends only on the abstract structure, instead of labellings or drawings of the graph. A chromatic polynomial is a polynomial that counts the number of graph colorings as a function of the number of colors. The Tutte polynomial is a two-variable polynomial on graph connectivity.
Geometric graph theory Geometric graph theory focuses on combinatorial and geometric properties of a graph that is drawn in a plane with straight-line or continuous curved edges in
Euclidean space. As part of
discrete geometry and
computational geometry, geometric graph theory studies
planar graphs, relationship to
higher-dimensional convex polytopes,
intersection of geometrical shaped sets, and other geometries' subareas of
incidence geometry and
projective geometry. A planar graph wherein its vertices are embedded as points, and its edges are non-crossing
line segments in the
Euclidean plane is called
planar straight-line graph. Any planar graph can be represented as a planar straight-line graph by
Fáry's theorem. The planar straight-line graph is the special case of a Euclidean graph. The Euclidean graph allows its edges to have the length of the
Euclidean distance between its endpoints. Its notions are the
Euclidean minimum spanning tree on minimizing the total length of the segments for finite points in any
Euclidean space,
Hadwiger–Nelson problem on asking for the minimum number of coloring plane such that no two points at a
unit distance from each other have the same color, and
shortest path problem on finding a path between two vertices in a graph that minimizes the sum of the assigned values of its edges. A
visibility graph is a graph whose vertices and edges are the point locations and visible connections, respectively. In a
simple polygon, where its edges are not self-intersecting and have no holes, the vertices of a visibility graph are connected by edges that represent the sides and diagonals of a polygon. The vertices are defined as the point locations.
Polyhedral graph is an undirected graph that forms the vertices and edges of a three-dimensional
convex polyhedron. In order to achieve it, such a graph must meet the requirements of
Steinitz's theorem, stating that every convex polyhedron is
3-vertex connected planar graph. The planar graph remains connected whenever any two of its vertices are removed. An
intersection graph is a graph in which each vertex is associated with a set and in which vertices are connected by edges whenever the corresponding sets have a nonempty
intersection. Each vertex is represented as a set, and every two vertices are connected. Hence, the intersection graph of finite sets can be represented through the smallest number of required elements, known as the
intersection number. The result graph can be geometric whenever the sets are geometrical objects. For instance, the intersection graph of line segments in one dimension is an
interval graph. The intersection graph of unit disks in the plane is a
unit disk graph. The intersection of a
circle packing is a
coin graph, where a vertex and an edge represent a circle and every pair of tangent circles; by Koebe–Andreev–Thurston theorem, the intersection graphs of non-crossing circles are exactly the planar graphs.
Scheinerman's theorem states that every planar graph can be represented as the intersection graph of line segments in the plane. The
Levi graph is a
bipartite graph that associates to the
incidence structure and
projective configuration. By applying to
information visualization, this creates another subarea of graph theory that is known as
graph drawing, which visualizes a graph depiction. Frequently drawn as node–link diagrams, the vertices of a graph are represented as disks, boxes, or textual labels, and the edges are represented as line segments, polylines, or curves in the Euclidean plane. Many definitions for graph drawings based on quality measures include the crossing number,
area, symmetry display on finding the problem of a graph's group automorphism,
bend minimization,
angular resolution, and
slope number. Tools for graph drawings are the circle packing, the intersection graph, and other visualizations of the adjacency matrix.
Extremal graph theory (illustrated), which is \lfloor n^2/4 \rfloor .
Extremal graph theory is a branch of mathematics at the intersection of
extremal combinatorics and graph theory. This area studies the maximum number of a graph's edges, known as the extremal number. The subarea's milestone originated from Mantel's theorem on the extremal number of a
triangle-free graph.
Turán's theorem extended Mantel's theorem for any undirected graph that does not have a complete subgraph of a given size. Turán's theorem is generalized by
Erdős–Stone theorem, which is occasionally known as the "fundamental theorem of extremal graph theory". Extremal graph theory also studies
forbidden subgraph problem,
homomorphism density, and
Szemerédi regularity lemma. The forbidden subgraphs problem suggests finding the extremal number of a graph with n vertices such that it does not have a subgraph that is
isomorphic to the graph. A
homomorphism density is a parameter that involves the
graph homomorphism. The density may refer to the probability that a map from the vertices of a graph to the vertices of another one chosen uniformly at random is a graph homomorphism. Being homomorphic means there exists a mapping between two graphs that respects their structure, or equivalently, a function between the vertex sets of two graphs that maps adjacent vertices to adjacent vertices. Szemerédi's regularity lemma states that a graph can be partitioned into a bounded number of parts so that the edges between parts are \varepsilon -regular.
Theory of random graph The theory of
random graph focuses on graphs using
probabilistic method. The subarea was founded by Hungarian mathematicians
Paul Erdős and
Alfréd Rényi, whose modelling generates random graphs, known as
Erdős–Rényi model. on the same graph but with randomized weights This subarea studies the
random tree. A
tree is an undirected graph where every pair of vertices is connected by exactly one
path. Therefore, a random tree is a tree that is formed by a
stochastic process. Many types of random trees include: •
Uniform spanning tree is a
spanning tree of a given graph in which each different tree is
equally likely to be selected. Being a spanning tree means that a subgraph is a tree that includes all of the vertices of a graph. The uniform spanning tree can be generated by using the method of random simple path known as the
loop-erased random walk, of taking a random walk on the given graph and erasing the cycles created by this walk. •
Branching process, a model of a population in which each individual has a random number of children •
Brownian tree, a fractal tree structure created by diffusion-limited aggregation processes •
Random binary tree is a
binary tree selected at random from some
probability distribution on binary trees. This includes trees formed by random insertion orders, and trees that are uniformly distributed with a given number of nodes. •
Random forest, a machine-learning classifier based on choosing random subsets of variables for each tree and using the most frequent tree output as the overall classification. •
Random minimal spanning tree is a spanning tree of a graph formed by choosing random edge weights and using the
minimum spanning tree for those weights. •
Random recursive tree, increasingly labelled trees, which can be generated using a simple stochastic growth rule. •
Rapidly exploring random tree, a fractal space-filling pattern used as a data structure for searching high-dimensional spaces. •
Treap or randomized binary search tree, a data structure that uses random choices to simulate a random binary tree for non-random update sequences
Graph enumeration == Applications ==