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Number of edges touching a vertex in a graph
identify a graph; in some cases, non-isomorphic graphs have the same degree sequence. A graph that is identified up to isomorphism by its degree sequence is called
Degree_(graph_theory)
Sequence of edges which join a sequence of vertices on a given graph
In graph theory, a path in a graph is a finite or infinite sequence of edges which joins a sequence of vertices which, by most definitions, are all distinct
Path_(graph_theory)
Graph in comparative genomics
Sequence graph, also called an alignment graph, breakpoint graph, or adjacency graph, are bidirected graphs used in comparative genomics. The structure
Sequence_graph
Graph with oriented edges
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
Directed_graph
Directed graph with no directed cycles
directed graph, each edge has an orientation, from one vertex to another vertex. A walk in a directed graph is a (finite or infinite) sequence ( v 1 ,
Directed_acyclic_graph
Directed graph representing overlaps between sequences of symbols
In graph theory, an n-dimensional De Bruijn graph of m symbols is a directed graph representing overlaps between sequences of symbols. It has mn vertices
De_Bruijn_graph
Class of artificial neural networks
Graph neural networks (GNNs) are artificial neural networks designed for tasks whose inputs are graphs. Because graphs usually do not have a canonical
Graph_neural_network
Property of graphs that depends only on abstract structure
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
Graph_property
Appendix:Glossary of graph theory in Wiktionary, the free dictionary. This is a glossary of graph theory. Graph theory is the study of graphs, systems of nodes
Glossary_of_graph_theory
Triangle-free graph requiring four colors
Grötzsch graph is sometimes also called the Mycielski graph or the Mycielski–Grötzsch graph. Unlike later graphs in this sequence, the Grötzsch graph is the
Grötzsch_graph
Pan-genome Graph Construction Methodology
pan-genome) of a species or a group of organisms. In such graphs, nodes often represent genomic sequences (e.g. DNA segments or k-mers) and edges represent adjacency
Pan-genome_graph_construction
Fast-growing function
satisfying the following: There is a sequence G 1 , … , G n {\displaystyle G_{1},\ldots ,G_{n}} of simple subcubic graphs such that each G i {\displaystyle
Friedman's_SSCG_function
Graph divided into two independent sets
In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets
Bipartite_graph
Graph representing edges of another graph
In the mathematical discipline of graph theory, the line graph of an undirected graph G is another graph L(G) that represents the adjacencies between edges
Line_graph
Structure on sequences of digits 1 and 2
Young–Fibonacci graph and Young–Fibonacci lattice, named after Alfred Young and Leonardo Fibonacci, are two closely related structures involving sequences of the
Young–Fibonacci_lattice
Graph in which every two vertices are adjacent
In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique
Complete_graph
Undirected, connected, and acyclic graph
In graph theory, a tree is an undirected graph in which every pair of distinct vertices is connected by exactly one path, or equivalently, a connected
Tree_(graph_theory)
Computer science algorithm
computer science, graph traversal (also known as graph search) refers to the process of visiting (checking and/or updating) each vertex in a graph. Such traversals
Graph_traversal
Graph which partitions into a clique and independent set
In graph theory, a branch of mathematics, a split graph is a graph in which the vertices can be partitioned into a clique and an independent set. Split
Split_graph
Count of crystal nodes by distance
In crystallography and graph theory, the coordination sequence of a vertex v {\displaystyle v} is an integer sequence that counts how many vertices are
Coordination_sequence
Type of knowledge base
knowledge graph is a knowledge base that uses a graph-structured data model or topology to represent and operate on data. Knowledge graphs are often used
Knowledge_graph
Trail in which only the first and last vertices are equal
(closed trail). Let G = (V, E, Φ) be a graph. A circuit is a non-empty trail (e1, e2, ..., en) with a vertex sequence (v1, v2, ..., vn, v1). A cycle or simple
Cycle_(graph_theory)
Fewest edge crossings in drawing of a graph
graph theory, the crossing number cr(G) of a graph G is the lowest number of edge crossings of a plane drawing of the graph G. For instance, a graph is
Crossing number (graph theory)
Crossing_number_(graph_theory)
Finite or infinite ordered list of elements
In mathematics, a sequence is a collection of objects possibly with repetition, that come in a specified order. Like a set, it contains members (also called
Sequence
Area of discrete mathematics
computer science, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context
Graph_theory
Graph where all long cycles have a chord
algorithm maintains a partition of the vertices of the graph into a sequence of sets; initially this sequence consists of a single set with all vertices. The
Chordal_graph
Function type in graph theory
important in the study of dense graphs. Graphons arise both as a natural notion for the limit of a sequence of dense graphs, and as the fundamental defining
Graphon
Basic concept of graph theory
mathematics and computer science, connectivity is one of the basic concepts of graph theory: it asks for the minimum number of elements (nodes or edges) that
Connectivity_(graph_theory)
Graphical representation of a computer program or algorithm
In computer science, a control-flow graph (CFG) is a representation, using graph notation, of all paths that might be traversed through a function during
Control-flow_graph
Partition of graph into sequence of paths
In graph theory, an ear of an undirected graph G is a path P where the two endpoints of the path may coincide, but where otherwise no repetition of edges
Ear_decomposition
Online database of integer sequences
links, and more, including the option to generate a graph or play a musical representation of the sequence. The database is searchable by keyword, by subsequence
On-Line Encyclopedia of Integer Sequences
On-Line_Encyclopedia_of_Integer_Sequences
Graph drawing with vertices in horizontal layers
construction of a layered graph drawing proceeds in a sequence of steps: If the input graph is not already a directed acyclic graph, a set of edges is identified
Layered_graph_drawing
Periodic spatial graph
Laves graph is an infinite and highly symmetric system of points and line segments in three-dimensional Euclidean space, forming a periodic graph. Three
Laves_graph
Mathematical sequence
times it appears in the sequence plus 1. For instance, in pseudo-code: Convert-Prüfer-to-Tree(a) 1 n ← length[a] 2 T ← a graph with n + 2 isolated nodes
Prüfer_sequence
Graph with a prism as its skeleton
prism graphs, and do not form a separate sequence of graphs. Prism graphs are examples of generalized Petersen graphs, with parameters GP(n,1). They may also
Prism_graph
Description of degree sequences of graphs
sufficient condition for a finite sequence of natural numbers to be the degree sequence of a simple graph. A sequence obeying these conditions is called
Erdős–Gallai_theorem
Graph representing intersections between given sets
In graph theory, an intersection graph is a graph that represents the pattern of intersections of a family of sets. Any graph can be represented as an
Intersection_graph
Cycle graph plus universal vertex
In graph theory, a wheel graph is a graph formed by connecting a single universal vertex to all vertices of a cycle. A wheel graph with n vertices can
Wheel_graph
Numbers obtained by adding the two previous ones
Fibonacci sequence is a sequence in which each element is the sum of the two elements that precede it. Numbers that are part of the Fibonacci sequence are known
Fibonacci_sequence
Assigning directions to the edges of an undirected graph
non-isomorphic oriented graphs with n vertices (for n = 1, 2, 3, …) is 1, 2, 7, 42, 582, 21480, 2142288, 575016219, 415939243032, … (sequence A001174 in the OEIS)
Orientation_(graph_theory)
Trail in a graph that visits each edge once
In graph theory, an Eulerian trail (or Eulerian path) is a trail in a finite graph that visits every edge exactly once (allowing for revisiting vertices)
Eulerian_path
Cubic graph with 10 vertices and 15 edges
mathematical field of graph theory, the Petersen graph is an undirected graph with 10 vertices and 15 edges. It is a small graph that serves as a useful
Petersen_graph
Algorithm in graph theory
there a simple graph such that its degree sequence is exactly this list? A simple graph contains no double edges or loops. The degree sequence is a list of
Havel–Hakimi_algorithm
Intersection graph of convex polygons whose vertices lie on a common circle
polygon-circle graph can be represented as an "alternating sequence". Such a sequence can be gained by perturbing the polygons representing the graph (if necessary)
Polygon-circle_graph
Mathematical graph relating to chess
In graph theory, a knight's graph, or a knight's tour graph, is a graph that represents all legal moves of the knight chess piece on a chessboard. Each
Knight's_graph
Graph with tight clique-coloring relation
monotonic sequences, can be expressed in terms of the perfection of certain associated graphs. The perfect graph theorem states that the complement graph of
Perfect_graph
Two closely related models for generating random graphs
Suppose that P is a graph property such that for every sequence M = M(n) with |M−(n2)p|=O(s(n)), the probability that a graph sampled from G(n, M) has
Erdős–Rényi_model
Subgraph with contracted edges
In graph theory, an undirected graph H is called a minor of the undirected graph G if H can be formed from G by deleting edges and vertices and by contracting
Graph_minor
Node ordering for directed acyclic graphs
application, a topological ordering is just a valid sequence for the tasks. Precisely, a topological sort is a graph traversal in which each node v is visited only
Topological_sorting
algorithm. Velvet: a set of algorithms manipulating de Bruijn graphs for genomic sequence assembly Geohash: a public domain algorithm that encodes a decimal
List_of_algorithms
Efficient version of non-strict evaluation
In computer science, graph reduction implements an efficient version of non-strict evaluation, an evaluation strategy where the arguments to a function
Graph_reduction
Graph with an antiprism as its skeleton
different sequence of graphs. An antiprism graph is a special case of a circulant graph, Ci2n(2,1). Other infinite sequences of polyhedral graph formed in
Antiprism_graph
Tree graph with one central node and leaves of length 1
be encoded by a Prüfer sequence; the Prüfer sequence for a star K1,k consists of k − 1 copies of the center vertex. Several graph invariants are defined
Star_(graph_theory)
Graph path which is an induced subgraph
mathematical area of graph theory, an induced path in an undirected graph G is a path that is an induced subgraph of G. That is, it is a sequence of vertices in
Induced_path
Intersection graph for intervals on the real number line
intersection graph of the intervals. Interval graphs are chordal graphs and perfect graphs. They can be recognized in linear time, and an optimal graph coloring
Interval_graph
Graph made from vertices and edges of a convex polyhedron
In geometric graph theory, a branch of mathematics, a polyhedral graph is the undirected graph formed from the vertices and edges of a convex polyhedron
Polyhedral_graph
Graph in which all ordered pairs of linked nodes are automorphic
vertices in the sequence are adjacent, and with any repeated vertices being more than 2 steps apart. A t-transitive graph is a graph such that the automorphism
Symmetric_graph
Cycle through all length-k sequences
(n − 1 = 4 − 1 = 3) 3D de Bruijn graph cycle. Each edge in this 3-dimensional de Bruijn graph corresponds to a sequence of four digits: the three digits
De_Bruijn_sequence
Directed graph where each vertex pair has one arc
In graph theory, a tournament is a directed graph with exactly one edge between each two vertices, in one of the two possible directions. Equivalently
Tournament_(graph_theory)
Generalization of graph theory
hypergraph is a generalization of a graph in which an edge can join any number of vertices. In contrast, in an ordinary graph, an edge connects exactly two
Hypergraph
Sloane, N. J. A. (ed.). "Sequence A000088 (Number of graphs on n unlabeled nodes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Cameron
Graph_enumeration
twin-width of an undirected graph is a natural number associated with the graph, used to study the parameterized complexity of graph algorithms. Intuitively
Twin-width
Natural number
In graph theory, all graphs with four or fewer vertices are planar, however, there is a graph with five vertices that is not: K5, the complete graph with
5
Recursively-formed graph with two terminal vertices
sink of Sc. A two-terminal series–parallel graph (TTSPG) is a graph that may be constructed by a sequence of series and parallel compositions starting
Series–parallel_graph
Partition of a graph whose components are reachable from all vertices
In the mathematical theory of directed graphs, a graph is said to be strongly connected if every vertex is reachable from every other vertex. The strongly
Strongly_connected_component
Software in bioinformatics
in the graph assembly. Nodes are built as (k-1)-mers connect by an edge. The assembler will then construct sequences based on the De Bruijn graph. De Bruijn
De_novo_sequence_assemblers
Data organization and storage formats
graph-based data structures are used in computer science and related fields: Graph Adjacency list Adjacency matrix Graph-structured stack Scene graph
List_of_data_structures
In graph theory, the Laman graphs are a family of sparse graphs describing the minimally rigid systems of rods and joints in the plane. Formally, a Laman
Laman_graph
Graph able to be partitioned into multiple independent sets
In graph theory, a part of mathematics, a k-partite graph is a graph whose vertices are (or can be) partitioned into k different independent sets. Equivalently
Multipartite_graph
problem is a classical decision problem in graph theory, a branch of combinatorics. Given two finite sequences ( a 1 , … , a n ) {\displaystyle (a_{1},\dots
Bipartite_realization_problem
Type of chart
A bar chart or bar graph is a chart or graph that presents categorical data with rectangular bars with heights or lengths proportional to the values that
Bar_chart
Graph generated by a random process
In mathematics, random graph is the general term to refer to probability distributions over graphs. Random graphs may be described simply by a probability
Random_graph
Constructs with triply-connected vertices
simple graphs are listed for small vertex numbers. The number of connected simple cubic graphs on 4, 6, 8, 10, ... vertices is 1, 2, 5, 19, ... (sequence A002851
Table_of_simple_cubic_graphs
Derived graph of higher chromatic number
one-edge graph, produces a sequence of graphs Mi = μ(Mi−1), sometimes called the Mycielski graphs. The first few graphs in this sequence are the graph M2 =
Mycielskian
Mode of convergence of a function sequence
functions of the sequence lie in a uniform error bar of the original function. Graphically this means that, given any thin band around the graph of f {\displaystyle
Uniform_convergence
Archimedean solid with 62 faces
is a zero-symmetric and cubic Archimedean graph. This polyhedron can be considered a member of a sequence of uniform patterns with vertex figure (4.6
Truncated_icosidodecahedron
Geometric graph with unit edge lengths
In mathematics, particularly geometric graph theory, a unit distance graph is a graph formed from a collection of points in the Euclidean plane by connecting
Unit_distance_graph
Type of graph
In graph theory, a biconnected graph is a connected and "nonseparable" graph, meaning that if any one vertex were to be removed, the graph will remain
Biconnected_graph
Existence of a graph with a degree sequence
The graph realization problem is a decision problem in graph theory. Given a finite sequence ( d 1 , … , d n ) {\displaystyle (d_{1},\dots ,d_{n})} of
Graph_realization_problem
Direction towards infinity in infinite graphs
sequence and each two consecutive vertices in the sequence are the two endpoints of an edge in the graph. According to Halin's definition, two rays r 0 {\displaystyle
End_(graph_theory)
combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, mathematical logic, number theory, set theory, Ramsey
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Unrelated vertices in graphs
In graph theory, an independent set, stable set, coclique or anticlique is a set of vertices in a graph, no two of which are adjacent. That is, it is a
Independent set (graph theory)
Independent_set_(graph_theory)
Theorem in graph theory
condition for two finite sequences of natural numbers to be the degree sequence of a labeled simple bipartite graph; a sequence obeying these conditions
Gale–Ryser_theorem
Directed graph where every node has exactly one path to it from the root
sequence: 0, 1, 1, 2, 4, 9, 20, 48, 115, 286, 719, 1842, 4766, 12486, ... (sequence A000081 in the OEIS). Edmonds' algorithm Multitree Rooted graph Darij
Arborescence_(graph_theory)
Method of DNA sequence reconstruction
bioinformatics, sequence assembly refers to aligning and merging fragments from a longer DNA sequence in order to reconstruct the original sequence. This is
Sequence_assembly
Open problem on 3x+1 and x/2 functions
34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1 . The sequence for n = 27, listed and graphed below, takes 111 steps (41 steps through odd numbers, in
Collatz_conjecture
Graph formed by complementation and disjoint union
In graph theory, a cograph, or complement-reducible graph, or P4-free graph, is a graph that can be generated from the single-vertex graph K1 by complementation
Cograph
Semicontinuity for set-valued functions
single-valued functions. To explain both notions, consider a sequence a of points in a domain, and a sequence b of points in the range. We say that b corresponds
Hemicontinuity
File format for graphs
GraphML is an XML-based file format for graphs. The GraphML file format results from the joint effort of the graph drawing community to define a common
GraphML
Balanced complete multipartite graph
The Turán graph, denoted by T ( n , r ) {\displaystyle T(n,r)} , is a complete multipartite graph; it is formed by partitioning a set of n {\displaystyle
Turán_graph
Class of graphs
In graph theory, a forcing graph is one whose density determines whether a graph sequence is quasi-random. The term was first coined by Chung, Graham,
Forcing_graph
Archimedean solid with 62 faces
pentagrammic prisms. In the mathematical field of graph theory, a rhombicosidodecahedral graph is the graph of vertices and edges of the rhombicosidodecahedron
Rhombicosidodecahedron
Cartesian product of complete graphs
positive integer. The Hamming graph H(d,q) has vertex set Sd, the set of ordered d-tuples of elements of S, or sequences of length d from S. Two vertices
Hamming_graph
Graph that displays observed data in a time sequencer
A run chart, also known as a run-sequence plot, is a graph that displays observed data in a time sequence. Often, the data displayed represent some aspect
Run_chart
Pattern of states and moves in the Tower of Hanoi puzzle
In graph theory and recreational mathematics, the Hanoi graphs are undirected graphs whose vertices represent the possible states of the Tower of Hanoi
Hanoi_graph
Tree graph with all nodes within distance 1 from central path
Hamiltonian graph. That is, in a caterpillar, there exists a cyclic sequence of all the vertices in which each adjacent pair of vertices in the sequence is at
Caterpillar_tree
On existence of a strongly regular graph
exist a strongly regular graph with parameters (99,14,1,2)? More unsolved problems in mathematics In graph theory, Conway's 99-graph problem is an unsolved
Conway's_99-graph_problem
Bipartite, 3-regular undirected graph
field of graph theory, the Pappus graph is a bipartite, 3-regular, undirected graph with 18 vertices and 27 edges, formed as the Levi graph of the Pappus
Pappus_graph
the update sequence is a permutation one frequently speaks of a permutation SDS to emphasize this point. Example: Let Y be the circle graph on vertices
Graph_dynamical_system
Graph of numbers differing by a square
Paley graphs form an infinite family of conference graphs, which yield an infinite family of symmetric conference matrices. Paley graphs allow graph-theoretic
Paley_graph
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