Consensus dynamics
Consensus dynamics or agreement dynamics is an area of research lying at the intersection of systems theory and graph theory. A major topic of investigation is the agreement or consensus problem in mu
Cycle decomposition (graph theory)
In graph theory, a cycle decomposition is a decomposition (a partitioning of a graph's edges) into cycles. Every vertex in a graph that has a cycle decomposition must have even degree.
Pósa's theorem
Pósa's theorem, in graph theory, is a sufficient condition for the existence of a Hamiltonian cycle based on the degrees of the vertices in an undirected graph. It implies two other degree-based suffi
Graph homomorphism
In the mathematical field of graph theory, a graph homomorphism is a mapping between two graphs that respects their structure. More concretely, it is a function between the vertex sets of two graphs t
Graph (discrete mathematics)
In discrete mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related". The objects corresp
GraphCrunch
GraphCrunch is a comprehensive, parallelizable, and easily extendible open source software tool for analyzing and modeling large biological networks (or graphs); it compares real-world networks agains
Hereditary property
In mathematics, a hereditary property is a property of an object that is inherited by all of its subobjects, where the meaning of subobject depends on the context. These properties are particularly co
Discharging method (discrete mathematics)
The discharging method is a technique used to prove lemmas in structural graph theory. Discharging is most well known for its central role in the proof of the four color theorem. The discharging metho
Convex subgraph
In metric graph theory, a convex subgraph of an undirected graph G is a subgraph that includes every shortest path in G between two of its vertices. Thus, it is analogous to the definition of a convex
Mediation-driven attachment model
In the scale-free network theory (mathematical theory of networks or graph theory), a mediation-driven attachment (MDA) model appears to embody a preferential attachment rule tacitly rather than expli
Graph property
In graph theory, a graph property or graph invariant is a property of graphs that depends only on the abstract structure, not on graph representations such as particular labellings or drawings of the
Copying network models
Copying network models are network generation models that use a copying mechanism to form a network, by repeatedly duplicating and mutating existing nodes of the network. Such a network model has firs
Graph theory
In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices (also called nod
Distance (graph theory)
In the mathematical field of graph theory, the distance between two vertices in a graph is the number of edges in a shortest path (also called a graph geodesic) connecting them. This is also known as
Discrete Laplace operator
In mathematics, the discrete Laplace operator is an analog of the continuous Laplace operator, defined so that it has meaning on a graph or a discrete grid. For the case of a finite-dimensional graph
Covering graph
In the mathematical discipline of graph theory, a graph C is a covering graph of another graph G if there is a covering map from the vertex set of C to the vertex set of G. A covering map f is a surje
Homomorphic equivalence
In the mathematics of graph theory, two graphs, G and H, are called homomorphically equivalent if there exists a graph homomorphism and a graph homomorphism . An example usage of this notion is that a
Random graph
In mathematics, random graph is the general term to refer to probability distributions over graphs. Random graphs may be described simply by a probability distribution, or by a random process which ge
Boundary (graph theory)
In graph theory, the outer boundary of a subgraph H of a graph G is the set of vertices of G not in H that have a common edge with a vertex in H. Its inner boundary is the set of vertices of H that ha
Degree distribution
In the study of graphs and networks, the degree of a node in a network is the number of connections it has to other nodes and the degree distribution is the probability distribution of these degrees o
Copying mechanism
In the study of scale-free networks, a copying mechanism is a process by which such a network can form and grow, by means of repeated steps in which nodes are duplicated with mutations from existing n
Pearls in Graph Theory
Pearls in Graph Theory: A Comprehensive Introduction is an undergraduate-level textbook on graph theory by and Gerhard Ringel. It was published in 1990 by Academic Press with a revised edition in 1994
Phase-field models on graphs
Phase-field models on graphs are a discrete analogue to phase-field models, defined on a graph. They are used in image analysis (for feature identification) and for the segmentation of social networks
Ryser's conjecture
In graph theory, Ryser's conjecture is a conjecture relating the maximum matching size and the minimum transversal size in hypergraphs. This conjecture first appeared in 1971 in the Ph.D. thesis of J.
Chip-firing game
The chip-firing game is a one-player game on a graph which was invented around 1983 and since has become an important part of the study of structural combinatorics.
Graph equation
In graph theory, Graph equations are equations in which the unknowns are graphs. One of the central questions of graph theory concerns the notion of isomorphism. We ask: When are two graphs the same?
Random graph theory of gelation
Random graph theory of gelation is a mathematical theory for sol–gel processes. The theory is a collection of results that generalise the Flory–Stockmayer theory, and allow identification of the gel p
Hydra game
In mathematics, specifically in graph theory and number theory, a hydra game is a single-player iterative mathematical game played on a mathematical tree called a hydra where, usually, the goal is to
The Mathematics of Chip-Firing
The Mathematics of Chip-Firing is a textbook in mathematics on chip-firing games and abelian sandpile models. It was written by Caroline Klivans, and published in 2018 by the CRC Press.
Vertex k-center problem
The vertex k-center problem is a classical NP-hard problem in computer science. It has application in facility location and clustering. Basically, the vertex k-center problem models the following real
Centrality
In graph theory and network analysis, indicators of centrality assign numbers or rankings to nodes within a graph corresponding to their network position. Applications include identifying the most inf
Structural induction
Structural induction is a proof method that is used in mathematical logic (e.g., in the proof of Łoś' theorem), computer science, graph theory, and some other mathematical fields. It is a generalizati
Weighted planar stochastic lattice
Physicists often use various lattices to apply their favorite models in them. For instance, the most favorite lattice is perhaps the square lattice. There are 14 Bravais space lattice where every cell
Homeomorphism (graph theory)
In graph theory, two graphs and are homeomorphic if there is a graph isomorphism from some of to some subdivision of . If the edges of a graph are thought of as lines drawn from one vertex to another
Capacitated arc routing problem
In mathematics, the capacitated arc routing problem (CARP) is that of finding the shortest tour with a minimum graph/travel distance of a mixed graph with undirected edges and directed arcs given capa
Deletion–contraction formula
In graph theory, a deletion-contraction formula / recursion is any formula of the following recursive form: Here G is a graph, f is a function on graphs, e is any edge of G, G \ e denotes edge deletio
Dominator (graph theory)
In computer science, a node d of a control-flow graph dominates a node n if every path from the entry node to n must go through d. Notationally, this is written as d dom n (or sometimes d ≫ n). By def
Incidence poset
In mathematics, an incidence poset or incidence order is a type of partially ordered set that represents the incidence relation between vertices and edges of an undirected graph. The incidence poset o
Maximally-matchable edge
In graph theory, a maximally-matchable edge in a graph is an edge that is included in at least one maximum-cardinality matching in the graph. An alternative term is allowed edge. A fundamental problem
Single-entry single-exit
In mathematics graph theory, a single-entry single-exit (SESE) region in a given graph is an ordered edge pair. For example, with the ordered edge pair, (a, b) of distinct control-flow edges a and b w
Burr–Erdős conjecture
In mathematics, the Burr–Erdős conjecture was a problem concerning the Ramsey number of sparse graphs. The conjecture is named after Stefan Burr and Paul Erdős, and is one of many conjectures named af
Markov chain
A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informa
Hierarchical closeness
Hierarchical closeness (HC) is a structural centrality measure used in network theory or graph theory. It is extended from closeness centrality to rank how centrally located a node is in a directed ne
The Petersen Graph
The Petersen Graph is a mathematics book about the Petersen graph and its applications in graph theory. It was written by Derek Holton and John Sheehan, and published in 1993 by the Cambridge Universi
Frequency partition of a graph
In graph theory, a discipline within mathematics, the frequency partition of a graph (simple graph) is a partition of its vertices grouped by their degree. For example, the degree sequence of the left
Calculus on finite weighted graphs
In mathematics, calculus on finite weighted graphs is a discrete calculus for functions whose domain is the vertex set of a graph with a finite number of vertices and weights associated to the edges.
Dot product representation of a graph
A dot product representation of a simple graph is a method of representing a graph using vector spaces and the dot product from linear algebra. Every graph has a dot product representation.
Null model
In mathematics, for example in the study of statistical properties of graphs, a null model is a type of random object that matches one specific object in some of its features, or more generally satisf
Icosian game
The icosian game is a mathematical game invented in 1857 by William Rowan Hamilton. The game's object is finding a Hamiltonian cycle along the edges of a dodecahedron such that every vertex is visited
Vertex cover in hypergraphs
In graph theory, a vertex cover in a hypergraph is a set of vertices, such that every hyperedge of the hypergraph contains at least one vertex of that set. It is an extension of the notion of vertex c
Graph removal lemma
In graph theory, the graph removal lemma states that when a graph contains few copies of a given subgraph, then all of the copies can be eliminated by removing a small number of edges.The special case
Nash-Williams theorem
In graph theory, the Nash-Williams theorem is a theorem that describes how many edge-disjoint spanning trees (and more generally forests) a graph can have: A graph G has t edge-disjoint spanning trees
Five room puzzle
This classical, popular puzzle involves a large rectangle divided into five "rooms". The objective of the puzzle is to cross each "wall" of the diagram with a continuous line only once.
Glossary of graph theory
This is a glossary of graph theory. Graph theory is the study of graphs, systems of nodes or vertices connected in pairs by lines or .
Graph Theory, 1736–1936
Graph Theory, 1736–1936 is a book in the history of mathematics on graph theory. It focuses on the foundational documents of the field, beginning with the 1736 paper of Leonhard Euler on the Seven Bri
Graph edit distance
In mathematics and computer science, graph edit distance (GED) is a measure of similarity (or dissimilarity) between two graphs.The concept of graph edit distance was first formalized mathematically b
Network theory
Network theory is the study of graphs as a representation of either symmetric relations or asymmetric relations between discrete objects. In computer science and network science, network theory is a p
Sidorenko's conjecture
Sidorenko's conjecture is a conjecture in the field of graph theory, posed by in 1986. Roughly speaking, the conjecture states that for any bipartite graph and graph on vertices with average degree ,
Graph (abstract data type)
In computer science, a graph is an abstract data type that is meant to implement the undirected graph and directed graph concepts from the field of graph theory within mathematics. A graph data struct
Graph dynamical system
In mathematics, the concept of graph dynamical systems can be used to capture a wide range of processes taking place on graphs or networks. A major theme in the mathematical and computational analysis
Node influence metric
In graph theory and network analysis, node influence metrics are measures that rank or quantify the influence of every node (also called vertex) within a graph. They are related to centrality indices.
Graphon
In graph theory and statistics, a graphon (also known as a graph limit) is a symmetric measurable function , that is important in the study of dense graphs. Graphons arise both as a natural notion for
Sequential dynamical system
Sequential dynamical systems (SDSs) are a class of graph dynamical systems. They are discrete dynamical systems which generalize many aspects of for example classical cellular automata, and they provi
Flag algebra
Flag algebras are an important computational tool in the field of graph theory which have a wide range of applications in homomorphism density and related topics. Roughly, they formalize the notion of
Large-scale capacitated arc routing probem
A large-scale capacitated arc routing problem (LSCARP) is a variant of the capacitated arc routing problem that covers 300 or more edges to model complex arc routing problems at large scales. Yi Mei e
Logic of graphs
In the mathematical fields of graph theory and finite model theory, the logic of graphs deals with formal specifications of graph properties using sentences of mathematical logic. There are several va
Kahn–Kalai conjecture
The Kahn–Kalai conjecture, also known as the expectation threshold conjecture, is a conjecture in the field of graph theory and statistical mechanics, proposed by Jeff Kahn and Gil Kalai in 2006.
Graph isomorphism
In graph theory, an isomorphism of graphs G and H is a bijection between the vertex sets of G and H such that any two vertices u and v of G are adjacent in G if and only if and are adjacent in H. This
Fractional graph isomorphism
In graph theory, a fractional isomorphism of graphs whose adjacency matrices are denoted A and B is a doubly stochastic matrix D such that DA = BD. If the doubly stochastic matrix is a permutation mat
Multi-trials technique
The multi-trials technique by Schneider et al. is employed for distributed algorithms and allows breaking of symmetry efficiently. Symmetry breaking is necessary, for instance, in resource allocation
Nullity (graph theory)
The nullity of a graph in the mathematical subject of graph theory can mean either of two unrelated numbers. If the graph has n vertices and m edges, then:
* In the matrix theory of graphs, the nulli
Quasirandom group
In mathematics, a quasirandom group is a group that does not contain a large product-free subset. Such groups are precisely those without a small non-trivial irreducible representation. The namesake o
Random cluster model
In statistical mechanics, probability theory, graph theory, etc. the random cluster model is a random graph that generalizes and unifies the Ising model, Potts model, and percolation model. It is used
Graph flattenability
Flattenability in some -dimensional normed vector space is a property of graphs which states that any embedding, or drawing, of the graph in some high dimension can be "flattened" down to live in -dim
Eigenvector centrality
In graph theory, eigenvector centrality (also called eigencentrality or prestige score) is a measure of the influence of a node in a network. Relative scores are assigned to all nodes in the network b
Icosian calculus
The icosian calculus is a non-commutative algebraic structure discovered by the Irish mathematician William Rowan Hamilton in 1856.In modern terms, he gave a group presentation of the icosahedral rota
Ramsey-Turán theory
Ramsey-Turán theory is a subfield of extremal graph theory. It studies common generalizations of Ramsey's theorem and Turán's theorem. In brief, Ramsey-Turán theory asks for the maximum number of edge
Implicit graph
In the study of graph algorithms, an implicit graph representation (or more simply implicit graph) is a graph whose vertices or edges are not represented as explicit objects in a computer's memory, bu
Graph entropy
In information theory, the graph entropy is a measure of the information rate achievable by communicating symbols over a channel in which certain pairs of values may be confused. This measure, first i
List of graph theory topics
This is a list of graph theory topics, by Wikipedia page. See glossary of graph theory terms for basic terminology
Ultragraph C*-algebra
In mathematics, an ultragraph C*-algebra is a universal C*-algebra generated by partial isometries on a collection of Hilbert spaces constructed from the ultragraphpg 6-7. These C*-algebras were creat
Distance oracle
In computing, a distance oracle (DO) is a data structure for calculating distances between vertices in a graph.
Graph Fourier transform
In mathematics, the graph Fourier transform is a mathematical transform which eigendecomposes the Laplacian matrix of a graph into eigenvalues and eigenvectors. Analogously to the classical Fourier Tr
Hall violator
In graph theory, a Hall violator is a set of vertices in a graph, that violate the condition to Hall's marriage theorem. Formally, given a bipartite graph G = (X + Y, E), a Hall-violator in X is a sub
Transitive reduction
In the mathematical field of graph theory, a transitive reduction of a directed graph D is another directed graph with the same vertices and as few edges as possible, such that for all pairs of vertic
Aanderaa–Karp–Rosenberg conjecture
In theoretical computer science, the Aanderaa–Karp–Rosenberg conjecture (also known as the Aanderaa–Rosenberg conjecture or the evasiveness conjecture) is a group of related conjectures about the numb
Degree (graph theory)
In graph theory, the degree (or valency) of a vertex of a graph is the number of edges that are incident to the vertex; in a multigraph, a loop contributes 2 to a vertex's degree, for the two ends of
Graph amalgamation
In graph theory, a graph amalgamation is a relationship between two graphs (one graph is an amalgamation of another). Similar relationships include subgraphs and minors. Amalgamations can provide a wa
Read's conjecture
Read's conjecture is a conjecture, first made by Ronald Read, about the unimodality of the coefficients of chromatic polynomials in the context of graph theory. In 1974, tightened this to the conjectu
Hypergraph removal lemma
In graph theory, the hypergraph removal lemma states that when a hypergraph contains few copies of a given sub-hypergraph, then all of the copies can be eliminated by removing a small number of hypere
Rainbow-independent set
In graph theory, a rainbow-independent set (ISR) is an independent set in a graph, in which each vertex has a different color. Formally, let G = (V, E) be a graph, and suppose vertex set V is partitio
Pseudorandom graph
In graph theory, a graph is said to be a pseudorandom graph if it obeys certain properties that random graphs obey with high probability. There is no concrete definition of graph pseudorandomness, but
Graph algebra
In mathematics, especially in the fields of universal algebra and graph theory, a graph algebra is a way of giving a directed graph an algebraic structure. It was introduced by McNulty and Shallon, an
Seven Bridges of Königsberg
The Seven Bridges of Königsberg is a historically notable problem in mathematics. Its negative resolution by Leonhard Euler in 1736 laid the foundations of graph theory and prefigured the idea of topo
Vertex (graph theory)
In discrete mathematics, and more specifically in graph theory, a vertex (plural vertices) or node is the fundamental unit of which graphs are formed: an undirected graph consists of a set of vertices
Friedman's SSCG function
In mathematics, a simple subcubic graph (SSCG) is a finite simple graph in which each vertex has degree at most three. Suppose we have a sequence of simple subcubic graphs G1, G2, ... such that each g
Hypergraph regularity method
In mathematics, the hypergraph regularity method is a powerful tool in extremal graph theory that refers to the combined application of the hypergraph regularity lemma and the associated counting lemm
Independence complex
The independence complex of a graph is a mathematical object describing the independent sets of the graph. Formally, the independence complex of an undirected graph G, denoted by I(G), is an abstract
Loop (graph theory)
In graph theory, a loop (also called a self-loop or a buckle) is an edge that connects a vertex to itself. A simple graph contains no loops. Depending on the context, a graph or a multigraph may be de
Forbidden graph characterization
In graph theory, a branch of mathematics, many important families of graphs can be described by a finite set of individual graphs that do not belong to the family and further exclude all graphs from t
Bicircular matroid
In the mathematical subject of matroid theory, the bicircular matroid of a graph G is the matroid B(G) whose points are the edges of G and whose independent sets are the edge sets of pseudoforests of
Graph coloring
In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest
Mixed graph
In graph theory, a mixed graph G = (V, E, A) is a graph consisting of a set of vertices V, a set of (undirected) edges E, and a set of directed edges (or arcs) A.
Magic graph
A magic graph is a graph whose edges are labelled by the first q positive integers, where q is the number of edges, so that the sum over the edges incident with any vertex is the same, independent of
Random walk closeness centrality
Random walk closeness centrality is a measure of centrality in a network, which describes the average speed with which randomly walking processes reach a node from other nodes of the network. It is si
Graph canonization
In graph theory, a branch of mathematics, graph canonization is the problem of finding a canonical form of a given graph G. A canonical form is a labeled graph Canon(G) that is isomorphic to G, such t
Friendship paradox
The friendship paradox is the phenomenon first observed by the sociologist Scott L. Feld in 1991 that most people have fewer friends than their friends have, on average. It can be explained as a form
Dense subgraph
In graph theory and computer science, a dense subgraph is a subgraph with a high level of internal connectivity. This is formalized as follows: let G = (V, E) be an undirected graph and let S = (VS, E
Blow-up lemma
The blow-up lemma, proved by János Komlós, Gábor N. Sárközy, and Endre Szemerédi in 1997, is an important result in extremal graph theory, particularly within the context of the regularity method. It
Meshulam's game
In graph theory, Meshulam's game is a game used to explain a theorem of Roy Meshulam related to the homological connectivity of the independence complex of a graph, which is the smallest index k such
Counting lemma
The counting lemmas this article discusses are statements in combinatorics and graph theory. The first one extracts information from -regular pairs of subsets of vertices in a graph , in order to guar
Directed graph
In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by directed edges, often called arcs.
Graph homology
In algebraic topology and graph theory, graph homology describes the homology groups of a graph, where the graph is considered as a topological space. It formalizes the idea of the number of "holes" i
Deficiency (graph theory)
Deficiency is a concept in graph theory that is used to refine various theorems related to perfect matching in graphs, such as Hall's marriage theorem. This was first studied by Øystein Ore. A related
Spatial network
A spatial network (sometimes also geometric graph) is a graph in which the vertices or edges are spatial elements associated with geometric objects, i.e., the nodes are located in a space equipped wit