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Index of articles associated with the same name
a graph polynomial is a graph invariant whose value is a polynomial. Invariants of this type are studied in algebraic graph theory. Important graph polynomials
Graph_polynomial
Function in algebraic graph theory
The chromatic polynomial is a graph polynomial studied in algebraic graph theory, a branch of mathematics. It counts the number of graph colorings as a
Chromatic_polynomial
Algebraic encoding of graph connectivity
The Tutte polynomial, also called the dichromate or the Tutte–Whitney polynomial, is a graph polynomial. It is a polynomial in two variables which plays
Tutte_polynomial
Type of mathematical expression
3) A polynomial function in one real variable can be represented by a graph. The graph of the zero polynomial f(x) = 0 is the x-axis. The graph of a degree
Polynomial
Polynomial whose roots are the eigenvalues of a matrix
characteristic polynomial to zero. In spectral graph theory, the characteristic polynomial of a graph is the characteristic polynomial of its adjacency
Characteristic_polynomial
Unsolved problem in computational complexity theory
computer science Can the graph isomorphism problem be solved in polynomial time? More unsolved problems in computer science The graph isomorphism problem is
Graph_isomorphism_problem
Graph polynomial generating numbers of matchings
the mathematical fields of graph theory and combinatorics, a matching polynomial (sometimes called an acyclic polynomial) is a generating function of
Matching_polynomial
Methodic assignment of colors to elements of a graph
In graph theory, graph coloring is a methodic assignment of labels traditionally called "colors" to elements of a graph. The assignment is subject to certain
Graph_coloring
Set of edges without common vertices
problems may be solved in polynomial time for bipartite graphs. Hall's marriage theorem provides a characterization of bipartite graphs which have a perfect
Matching_(graph_theory)
Kauffman polynomial Graph polynomial, a similar class of polynomial invariants in graph theory Tutte polynomial, a special type of graph polynomial related
Knot_polynomial
Area of discrete mathematics
Algebraic graph theory also studies the algebraic invariants, chromatic polynomial, Tutte polynomial of a graph, and knot invariant. A graph invariant
Graph_theory
Subgraph with contracted edges
edge contractions. For every fixed graph H, it is possible to test whether H is a minor of an input graph G in polynomial time; together with the forbidden
Graph_minor
Graph where all long cycles have a chord
polynomial time when the input is chordal. The treewidth of an arbitrary graph may be characterized by the size of the cliques in the chordal graphs that
Chordal_graph
Linear map or polynomial function of degree one
function whose graph is a straight line, that is, a polynomial function of degree zero (a constant polynomial) or one (a linear polynomial). For distinguishing
Linear_function
Complexity class
computed in polynomial time, but finding the optimal solution is NP-complete. An interesting example is the graph isomorphism problem, the graph theory problem
NP-completeness
Unrelated vertices in graphs
set may be found in polynomial time. Famous examples are claw-free graphs, P5-free graphs and perfect graphs. For chordal graphs, a maximum weight independent
Independent set (graph theory)
Independent_set_(graph_theory)
Computational complexity class
colored directed graph. The paper giving a quasi-polynomial algorithm for these games won the 2021 Nerode Prize. 3-coloring circle graphs. These are the
Quasi-polynomial_time
Linear algebra aspects of graph theory
mathematics, spectral graph theory is the study of the properties of a graph in relationship to the characteristic polynomial, eigenvalues, and eigenvectors
Spectral_graph_theory
Graph with tight clique-coloring relation
all perfect graphs, the graph coloring problem, maximum clique problem, and maximum independent set problem can all be solved in polynomial time, despite
Perfect_graph
Bijection between the vertex set of two graphs
is called an isomorphism class of graphs. The question of whether graph isomorphism can be determined in polynomial time is a major unsolved problem in
Graph_isomorphism
Estimate of time taken for running an algorithm
clique and a random graph. Although quasi-polynomially solvable, it has been conjectured that the planted clique problem has no polynomial time solution; this
Time_complexity
Polynomial function of degree two
terms quadratic function and quadratic polynomial are nearly synonymous and often abbreviated as quadratic. The graph of a real single-variable quadratic
Quadratic_function
Branch of mathematics
The chromatic polynomial of a graph, for example, counts the number of its proper vertex colorings. For the Petersen graph, this polynomial is t ( t − 1
Algebraic_graph_theory
Unsolved problem in computer science
the solution to a problem can be checked in polynomial time, must the problem be solvable in polynomial time? More unsolved problems in computer science
P_versus_NP_problem
Structure-preserving correspondence between node-link graphs
homomorphism between given graphs is prohibitive in general, but a lot is known about special cases that are solvable in polynomial time. Boundaries between
Graph_homomorphism
List of unsolved computational problems
polynomial time on a classical or quantum computer? Can the graph isomorphism problem be solved in polynomial time on a classical computer? The graph
List of unsolved problems in computer science
List_of_unsolved_problems_in_computer_science
Directed graph with no directed cycles
In mathematics, particularly graph theory and computer science, a directed acyclic graph (DAG) is a directed graph with no directed cycles. That is, it
Directed_acyclic_graph
Point where function's value is zero
root of a polynomial is a zero of the corresponding polynomial function. The fundamental theorem of algebra shows that any non-zero polynomial has a number
Zero_of_a_function
Method for solving one problem using another
In computational complexity theory, a polynomial-time reduction is a method for solving one problem using another. One shows that if a hypothetical subroutine
Polynomial-time_reduction
Formula in graph theory
chromatic polynomial is one such function, and Tutte began to discover more, including a function f = t(G) counting the number of spanning trees of a graph (also
Deletion–contraction_formula
Mapping a graph onto itself without changing edge-vertex connectivity
connected graph – indeed, of a cubic graph. Constructing the automorphism group of a graph, in the form of a list of generators, is polynomial-time equivalent
Graph_automorphism
Graph able to be partitioned into multiple independent sets
the tripartite graphs. Bipartite graphs may be recognized in polynomial time but, for any k > 2 it is NP-complete, given an uncolored graph, to test whether
Multipartite_graph
Algorithm for division of polynomials
In algebra, polynomial long division is an algorithm for dividing a polynomial by another polynomial of the same or lower degree, a generalized version
Polynomial_long_division
Complexity class used to classify decision problems
repeatedly (a polynomial number of times). The subgraph isomorphism problem of determining whether graph G contains a subgraph that is isomorphic to graph H. Turing
NP_(complexity)
Problem of finding the longest simple path for a given graph
weighted graphs) by the sum of the weights of its edges. In contrast to the shortest path problem, which can be solved in polynomial time in graphs without
Longest_path_problem
Polynomial function of degree 3
may not be distinct); all odd-degree polynomials with real coefficients have at least one real root. The graph of a cubic function always has a single
Cubic_function
Property of graphs that depends only on abstract structure
path graph on 4 vertices both have the same chromatic polynomial, for example. Connected graphs Bipartite graphs Planar graphs Triangle-free graphs Perfect
Graph_property
Hungarian mathematician (born 1943)
Bollobás has proved results on extremal graph theory, functional analysis, the theory of random graphs, graph polynomials and percolation. For example, with
Béla_Bollobás
Pair of polynomial sequences
The Chebyshev polynomials are two sequences of orthogonal polynomials related to the cosine and sine functions, notated as T n ( x ) {\displaystyle T_{n}(x)}
Chebyshev_polynomials
which operates by drawing the graph of the polynomial on a plane and find the roots as the intersections of the graph with x-axis. In 1770, the English
Polynomial_root-finding
Spectral graph theory concept
spectral graph theory, a Ramanujan graph is a regular graph whose spectral gap is almost as large as possible (see extremal graph theory). Such graphs are
Ramanujan_graph
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
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
Cubic graph with 10 vertices and 15 edges
bridgeless graph has a cycle-continuous mapping to the Petersen graph. More unsolved problems in mathematics In the mathematical field of graph theory, the
Petersen_graph
System of complete and orthogonal polynomials
mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a wide number of
Legendre_polynomials
Polynomials used for interpolation
In numerical analysis, the Lagrange interpolating polynomial is the unique polynomial of lowest degree that interpolates a given set of data. Given a
Lagrange_polynomial
Representation of a mathematical function
\{a,b,c,d\}} , however, cannot be determined from the graph alone. The graph of the cubic polynomial on the real line f ( x ) = x 3 − 9 x {\displaystyle
Graph_of_a_function
Logical formulation of graph properties
property, with polynomial delay (as a function of n {\displaystyle n} ) per graph. A similar analysis can be performed for non-uniform random graphs, where the
Logic_of_graphs
Tree which includes all vertices of a graph
embedding can be found in polynomial time. A tree is a connected undirected graph with no cycles. It is a spanning tree of a graph G if it spans G (that is
Spanning_tree
On short connecting nets with added points
polynomial time. Despite the pessimistic worst-case complexity, several Steiner tree problem variants, including the Steiner tree problem in graphs and
Steiner_tree_problem
Task in computational graph theory
sometimes known as graph canonicalization. Unsolved problem in computer science Is graph canonization polynomial-time equivalent to the graph isomorphism problem
Graph_canonization
Task of computing complete subgraphs
families of graphs in which the number of cliques is polynomially bounded. These families include chordal graphs, complete graphs, triangle-free graphs, interval
Clique_problem
Graph representing faces of another graph
mathematical discipline of graph theory, the dual graph of a planar graph G is a graph that has a vertex for each face of G. The dual graph has an edge for each
Dual_graph
Maximal subgraph whose vertices can reach each other
the chromatic polynomial of the graph, and the chromatic polynomial of the whole graph can be obtained as the product of the polynomials of its components
Component_(graph_theory)
On the number of spanning trees in a graph
of the graph's Laplacian matrix. This shows in particular that the number of spanning trees can be computed from the graph data in polynomial time. Kirchhoff's
Kirchhoff's_theorem
In graph theory, a class of graphs is said to have few cliques if every member of the class has a polynomial number of maximal cliques. Certain generally
Graphs_with_few_cliques
Intersection graph of a chord diagram
general graphs have polynomial time algorithms when restricted to circle graphs. For instance, Kloks (1996) showed that the treewidth of a circle graph can
Circle_graph
Combinatorial optimization problem
problem, both parts of the bipartite graph have the same number of vertices, denoted by n. One of the first polynomial-time algorithms for balanced assignment
Assignment_problem
(2012). "Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs". Journal of the American Mathematical Society. 25 (3): 907–927.
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Abstraction of linear independence of vectors
an evaluation of the Tutte polynomial. The Tutte polynomial T G {\displaystyle T_{G}} of a graph is the Tutte polynomial T M ( G ) {\displaystyle T_{M(G)}}
Matroid
Field of electrical engineering
analyzed using linear methods. Polynomial signal processing is a type of non-linear signal processing, where polynomial systems may be interpreted as conceptually
Signal_processing
Graph drawing used to study Riemann surfaces
{\displaystyle d} white leaves (a complete bipartite graph K 1 , d {\displaystyle K_{1,d}} ). More generally, a polynomial p ( x ) {\displaystyle p(x)} having two
Dessin_d'enfant
Family of graphs whose shallow minors are sparse graphs
Many natural families of sparse graphs have bounded expansion. A closely related but stronger property, polynomial expansion, is equivalent to the existence
Bounded_expansion
Class of problems solvable in polynomial time
solved by a deterministic Turing machine using a polynomial amount of computation time, or polynomial time. Cobham's thesis holds that P is the class of
P_(complexity)
Path in a graph that visits each vertex exactly once
the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected or directed graph that visits each vertex exactly
Hamiltonian_path
Triangle-free graph requiring four colors
their distance from the degree-5 vertex. The characteristic polynomial of the Grötzsch graph is ( x − 1 ) 5 ( x 2 − x − 10 ) ( x 2 + 3 x + 1 ) 2 . {\displaystyle
Grötzsch_graph
In mathematics, and, in particular, in graph theory, a rooted graph is a graph in which one vertex has been distinguished as the root. Both directed and
Rooted_graph
Embedding of the circle in three dimensional Euclidean space
mathematics that studies knots is known as knot theory and has many relations to graph theory. A knot is an embedding of the circle (S1) into three-dimensional
Knot_(mathematics)
Measure of capacity of a communications channel defined from a graph
communications channel defined from the graph, and is upper bounded by the Lovász number, which can be computed in polynomial time. However, the computational
Shannon_capacity_of_a_graph
Graph with directed and undirected edges
our graph as the chromatic number, denoted by χ(G). The number of proper k-colorings is a polynomial function of k called the chromatic polynomial of our
Mixed_graph
Graph that can be embedded in the plane
In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect
Planar_graph
Finiteness of sets of forbidden graph minors
graph theory, the Robertson–Seymour theorem (also called the graph minors theorem) states that the undirected graphs, partially ordered by the graph minor
Robertson–Seymour_theorem
Cubic graph with 8 vertices and 12 edges
including both rotations and reflections. The characteristic polynomial of the Wagner graph is ( x − 3 ) ( x − 1 ) 2 ( x + 1 ) ( x 2 + 2 x − 1 ) 2 . {\displaystyle
Wagner_graph
Mathematical expression
Newton polynomial, named after its inventor Isaac Newton, is an interpolation polynomial for a given set of data points. The Newton polynomial is sometimes
Newton_polynomial
Mathematical function defined piecewise by polynomials
function defined piecewise by polynomials. In interpolating problems, spline interpolation is often preferred to polynomial interpolation because it yields
Spline_(mathematics)
Complexity class
every problem L which can be solved in non-deterministic polynomial-time, there is a polynomial-time reduction from L to H. That is, assuming a solution
NP-hardness
2-vertex-connected graph and a 2-edge-connected graph. Its chromatic polynomial is ( x − 2 ) ( x − 1 ) x . {\displaystyle (x-2)(x-1)x.} Triangle-free graph Weisstein
Triangle_graph
Subdivision of vertices into disjoint sets
components, it can be shown that no reasonable fully polynomial algorithms exist for these graphs. Consider a graph G = (V, E), where V denotes the set of n vertices
Graph_partition
Bollobás–Riordan polynomial can mean a 3-variable invariant polynomial of graphs on orientable surfaces, or a more general 4-variable invariant of ribbon graphs, generalizing
Bollobás–Riordan_polynomial
Partition of a graph's nodes into 2 disjoint subsets
In graph theory, a cut is a partition of the vertices of a graph into two disjoint subsets. Any cut determines a cut-set, the set of edges that have one
Cut_(graph_theory)
Number of times an object must be counted for making true a general formula
derivative. The discriminant of a polynomial is zero if and only if the polynomial has a multiple root. The graph of a polynomial function f intersects the x-axis
Multiplicity_(mathematics)
Edge-face adjacencies in another graph
graph theory, the medial graph of plane graph G is another graph M(G) that represents the adjacencies between edges in the faces of G. Medial graphs were
Medial_graph
Cycle graph plus universal vertex
W7 is the only wheel graph that is a unit distance graph in the Euclidean plane. The chromatic polynomial of the wheel graph Wn is : P W n ( x ) = x
Wheel_graph
Distance-regular graph with 56 vertices
Gosset graph, E7, acts transitively upon its vertices, making it a vertex-transitive graph. The characteristic polynomial of the Gosset graph is ( x −
Gosset_graph
Theory of getting acceptably close inexact mathematical calculations
contrived functions f(x) for which no such polynomial exists, but these occur rarely in practice. For example, the graphs shown to the right show the error in
Approximation_theory
Number of vertices with unambiguous distances
cographs, chain graphs, and cactus block graphs (a class including both cactus graphs and block graphs). The problem may be solved in polynomial time on outerplanar
Metric dimension (graph theory)
Metric_dimension_(graph_theory)
Graph with all vertices of degree 3
of graph theory, a cubic graph is a graph in which all vertices have degree three. In other words, a cubic graph is a 3-regular graph. Cubic graphs are
Cubic_graph
Sequence of differential equation solutions
In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are nontrivial solutions of Laguerre's differential equation: x y ″
Laguerre_polynomials
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
Square matrix used to represent a graph or network
In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether
Adjacency_matrix
Bivariegated graph Cage (graph theory) Cayley graph Circle graph Clique graph Cograph Common graph Complement of a graph Complete graph Cubic graph Cycle graph De
List_of_graph_theory_topics
Planar, undirected graph with 2n vertices and 3n-2 edges
chromatic index 3 (if n>2). The chromatic number of the ladder graph is 2 and its chromatic polynomial is ( x − 1 ) x ( x 2 − 3 x + 3 ) ( n − 1 ) {\displaystyle
Ladder_graph
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
Formula that provides the solutions to a quadratic equation
( x , y ) {\displaystyle (x,y)} -coordinates are the graph of a second-degree polynomial, of the form y = a x 2 + b x + c {\displaystyle \textstyle
Quadratic_formula
Inherent difficulty of computational problems
If graph isomorphism is NP-complete, the polynomial time hierarchy collapses to its second level. Since it is widely believed that the polynomial hierarchy
Computational complexity theory
Computational_complexity_theory
Graph representing a permutation
a permutation graph is polynomial in the size of the graph. Permutation graphs are a special case of circle graphs, comparability graphs, the complements
Permutation_graph
One of two different regular graphs with 16 vertices
Clebsch graph is an integral graph: its spectrum consists entirely of integers. The Clebsch graph is the only graph with this characteristic polynomial, making
Clebsch_graph
Graph with sign-labeled edges
In the area of graph theory in mathematics, a signed graph is a graph in which each edge has a positive or negative sign. A signed graph is balanced if
Signed_graph
Polynomial function of degree at most one
functions. With this definition, the degree of a linear polynomial would be exactly one, and its graph would be a line that is neither vertical nor horizontal
Linear_function_(calculus)
Planar graph with 4 nodes and 5 edges
characteristic polynomial of the diamond graph is x ( x + 1 ) ( x 2 − x − 4 ) {\displaystyle x(x+1)(x^{2}-x-4)} . It is the only graph with this characteristic
Diamond_graph
GRAPH POLYNOMIAL
GRAPH POLYNOMIAL
Boy/Male
Indian
Grape
Boy/Male
Biblical
A grape, a knot.
Girl/Female
Muslim
Grape like
Girl/Female
Muslim
Grape vine
Biblical
a grape; a knot
Girl/Female
Indian
Grape vine
Girl/Female
Arabic, Assamese, Hindu, Indian, Kannada, Malayalam, Marathi, Muslim, Telugu
Grape
Boy/Male
African, Arabic
Grape Vines
Boy/Male
Afghan, Hebrew, Indian, Parsi, Sanskrit
Grape Presser; World; Song
Boy/Male
Hebrew, Hindu, Indian, Marathi
Grape Cluster
Boy/Male
Hindu, Indian
Efficient; Conqueror of Miseries; Bond in Affection; Capable; Mysterious; Different than Others; Smart; Most Mysterious Vastu Grah 'Rahu'; Son of Lord Buddha; Son of Goddess Durga; Truth Follower; Best of All
Girl/Female
Afghan, Arabic, Hebrew, Indian, Muslim, Parsi, Sanskrit
Grape Presser; World; Song; Universe
Boy/Male
Biblical
A grape, a knot.
Female
Thai/Siamese
Thai name A-GUN means "grape."
Girl/Female
Indian
Grape like
Girl/Female
Hindu
Grape, Belonging to kashmir
Boy/Male
Muslim
Grape
Boy/Male
Arabic, Modern
Grape
Girl/Female
Tamil
Kaslunira | கஸà¯à®²à¯à®‚நீரா
Grape, Belonging to kashmir
Kaslunira | கஸà¯à®²à¯à®‚நீரா
Boy/Male
Hindu, Indian, Punjabi, Sikh
From Kashmir; Grape
GRAPH POLYNOMIAL
GRAPH POLYNOMIAL
Boy/Male
Tamil
Lord Vishnu
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Tamil, Telugu
Sun; Mine of Nectar; The Moon
Girl/Female
Latin
Little precious jewel.
Girl/Female
Arabic, Muslim
Delight; Joy
Boy/Male
Hindu, Indian
Sun Power in
Girl/Female
Greek Latin
A witch.
Boy/Male
Hebrew
Beloved by God.
Girl/Female
Bengali, Hindu, Indian, Kannada, Malayalam, Marathi, Tamil, Telugu
Bestower of Longevity
Boy/Male
Hindu
One of the kauravas, Arjuna
Girl/Female
Hindu, Indian
Lord Vishnu
GRAPH POLYNOMIAL
GRAPH POLYNOMIAL
GRAPH POLYNOMIAL
GRAPH POLYNOMIAL
GRAPH POLYNOMIAL
n.
A grape, or a bunch of grapes.
a.
Resembling a grape.
n.
A seed of the grape.
n.
A white grape, esteemed for the table.
n.
A mangy tumor on the leg of a horse.
a.
Composed of, or resembling, grapes.
n.
A grape dried in the sun; a raisin.
n.
A well-known edible berry growing in pendent clusters or bunches on the grapevine. The berries are smooth-skinned, have a juicy pulp, and are cultivated in great quantities for table use and for making wine and raisins.
n.
A plant of the genus Muscari; grape hyacinth.
n.
The plant which bears this fruit; the grapevine.
n.
A variety of shaddock, called also grape fruit.
n.
A grape of many varieties and colors.
a.
Full of small kernels like a grape.
n.
The cultivation of the vine; grape growing.
n.
The Hartford grape, a variety of grape first raised at Hartford, Connecticut, from the Northern fox grape. Its large dark-colored berries ripen earlier than those of most other kinds.
n.
See Grasshopper, and Frog hopper, Grape hopper, Leaf hopper, Tree hopper, under Frog, Grape, Leaf, and Tree.
n.
Grapeshot.
n.
A sort of grape.