* Enumerative Combinatorics by George E. Martin, ISBN 0-387-95225-X

The book Enumerative Combinatorics: Volume 2 by combinatorialist Richard P. Stanley contains a set of exercises which describe 66 different interpretations of the Catalan numbers.

* " Approaches to the Enumerative Theory of Meanders " by Michael La Croix

The notation S ( n, k ) was used by Richard Stanley in his book Enumerative Combinatorics.

Enumerative combinatorics is the most classical area of combinatorics, and concentrates on counting the number of certain combinatorial objects.

Enumerative definitions are only possible for finite sets and only practical for relatively small sets.

Enumerative bibliographies are based on a unifying principle such as creator, subject, date, topic or other characteristic.

* 2001 Richard P. Stanley, for his two-volume Enumerative Combinatorics

Enumerative Combinatorics, Vol.

Enumerative geometry remained his focus up until 1875.

* In graph theory an automorphism of a graph is a permutation of the nodes that preserves edges and non-edges.

There are also connections to string theory, game theory, graph matchings, solitons and integer programming.

Binary relations are used in many branches of mathematics to model concepts like " is greater than ", " is equal to ", and " divides " in arithmetic, " is congruent to " in geometry, " is adjacent to " in graph theory, " is orthogonal to " in linear algebra and many more.

Some mathematicians, especially in set theory, do not consider the sets and to be part of the relation, and therefore define a binary relation as being a subset of x, that is, just the graph.

One of the oldest and most accessible parts of combinatorics is graph theory, which also has numerous natural connections to other areas.

Five binary tree s on three Vertex ( graph theory ) | vertices, an example of Catalan number s.

It should be noted that while there are very strong connections between graph theory and combinatorics, these two are sometimes thought of as separate subjects.

* Chord ( graph theory ), an edge joining two not-adjacent nodes in a cycle

* Conjugate ( graph theory ), an alternative term for a line graph

* Digraph or directed graph, in graph theory

Much research in graph theory was motivated by attempts to prove that all maps, like this one, could be graph coloring | colored with four color theorem | only four colors.

In graph theory, much research was motivated by attempts to prove the four color theorem, first stated in 1852, but not proved until 1976 ( by Kenneth Appel and Wolfgang Haken, using substantial computer assistance ).

The telecommunication industry has also motivated advances in discrete mathematics, particularly in graph theory and information theory.

Several fields of discrete mathematics, particularly theoretical computer science, graph theory, and combinatorics, are important in addressing the challenging bioinformatics problems associated with understanding the tree of life.

It draws heavily on graph theory and logic.

* Eccentricity ( graph theory ) of a vertex in a graph

In graph theory, the term fullerene refers to any 3-regular, planar graph with all faces of size 5 or 6 ( including the external face ).

If the function f is not linear ( i. e. its graph is not a straight line ), however, then the change in y divided by the change in x varies: differentiation is a method to find an exact value for this rate of change at any given value of x.

If f is a continuous function, meaning that its graph is an unbroken curve with no gaps, then Q is a continuous function away from.

If the limit exists, meaning that there is a way of choosing a value for Q ( 0 ) that makes the graph of Q a continuous function, then the function f is differentiable at a, and its derivative at a equals Q ( 0 ).

If there is a graph requiring 5 colors, then there is a minimal such graph, where removing any vertex makes it four-colorable.

Call this graph G. G cannot have a vertex of degree 3 or less, because if d ( v ) ≤ 3, we can remove v from G, four-color the smaller graph, then add back v and extend the four-coloring to it by choosing a color different from its neighbors.

For example, the case described in degree 4 vertex situation is the configuration consisting of a single vertex labelled as having degree 4 in G. As above, it suffices to demonstrate that if the configuration is removed and the remaining graph four-colored, then the coloring can be modified in such a way that when the configuration is re-added, the four-coloring can be extended to it as well.

To prove this, one can combine a proof of the theorem for finite planar graphs with the De Bruijn – Erdős theorem stating that, if every finite subgraph of an infinite graph is k-colorable, then the whole graph is also k-colorable.

If the graph G is connected, then the rank of the free group is equal to 1 − χ ( G ): one minus the Euler characteristic of G.

If A is nonzero, then the x-intercept, that is, the x-coordinate of the point where the graph crosses the x-axis ( where, y is zero ), is − C / A.

If B is nonzero, then the y-intercept, that is the y-coordinate of the point where the graph crosses the y-axis ( where x is zero ), is − C / B, and the slope of the line is − A / B.

# Structured prediction: When the desired output value is a complex object, such as a parse tree or a labeled graph, then standard methods must be extended.

Each object that is referenced by the serialized object and not marked as < tt > transient </ tt > must also be serialized ; and if any object in the complete graph of non-transient object references is not serializable, then serialization will fail.

The Laffer curve is typically represented as a graph which starts at 0 % tax, zero revenue, rises to a maximum rate of revenue raised at an intermediate rate of taxation and then falls again to zero revenue at a 100 % tax rate.

; Global value numbering: GVN eliminates redundancy by constructing a value graph of the program, and then determining which values are computed by equivalent expressions.

This diagram gives us an idea about decoding: if a received sequence doesn't fit this graph, then it was received with errors, and we must choose the nearest correct ( fitting the graph ) sequence.

If it is constrained to bury the cable only along certain paths, then there would be a graph representing which points are connected by those paths.

There may be several minimum spanning trees of the same weight having a minimum number of edges ; in particular, if all the edge weights of a given graph are the same, then every spanning tree of that graph is minimum.

If there are n vertices in the graph, then each tree has n-1 edges.

For instance, if I is the category of the directed graph, then C < sup > I </ sup > has as objects the morphisms of C, and a morphism between and in C < sup > I </ sup > is a pair of morphisms and in C such that the " square commutes ", i. e..