The graph, as a set, may have a finite number of components.

If the force required to remove the coatings is plotted against film thickness, a graph as illustrated schematically in Fig. 5 may characteristically result.

According to the definition above, two relations with the same graph may be different, if they differ in the sets and.

It may be represented as a table or graph relating price and quantity supplied.

One of the most famous and productive problems of graph theory is the four color problem: " Is it true that any map drawn in the plane may have its regions colored with four colors, in such a way that any two regions having a common border have different colors?

( N. B., while hierarchies are commonly studied using graph theory, the general terminology used is different, and words such as " direct " may have different general meanings )

For instance, if one graphs a set of solutions of an equation in some higher dimensional space, he may ask about the geometric properties of the graph.

That algorithm enforces two constraints: children precede their parents and if a class inherits from multiple classes, they are kept in the order specified in the tuple of base classes ( however in this case, some classes high in the inheritance graph may precede classes lower in the graph ).

The goal of this problem is to minimize the cost of reaching a target in a weighted graph where some of the edges are unreliable and may have been removed from the graph.

* A random graph, for a graph with V edges, may be parameterized as an NxN matrix, indicating the weight for each edge, or 0 for no edge.

The items may be stored individually as records in a database ; or may be elements of a search space defined by a mathematical formula or procedure, such as the roots of an equation with integer variables ; or a combination of the two, such as the Hamiltonian circuits of a graph.

A point-like particle's motion may be described by drawing a graph of its position ( in one or two dimensions of space ) against time.

In addition to explaining the forces listed in the graph, a ToE may also explain the status of at least two candidate forces suggested by modern cosmology: an inflationary force and dark energy.

In the asymmetric TSP, paths may not exist in both directions or the distances might be different, forming a directed graph.

Namely, any free group G may be realized as the fundamental group of a graph X.

By the use of various weights, data for a force-rate of shear graph can be obtained.

Then every component of the graph of F must be defined over a bounded sub-interval.

The resolution has to be equal to or better than the half width of an atomic absorption line ( about 2 pm ) in order to avoid losses of sensitivity and linearity of the calibration graph.

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.

Strings-and-coins can be played on an arbitrary graph.

Many other decision problems, such as graph coloring problems, planning problems, and scheduling problems, can be easily encoded into SAT.

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.

A function from the set of real numbers to the real numbers can be represented by a graph in the Cartesian plane ; the function is continuous if, roughly speaking, the graph is a single unbroken curve with no " holes " or " jumps ".

Graphing calculators can be used to graph functions defined on the real line, or higher dimensional Euclidean space.

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.

It is worth noting that the vertices and edges of a convex polyhedron can be projected to form a graph ( sometimes called a Schlegel diagram ) on the sphere or on a flat plane, and the corresponding graph formed by the dual of this polyhedron is its dual graph.

** In power engineering, a " bus " is any graph node of the single-line diagram at which voltage, current, power flow, or other quantities are to be evaluated.

Thus, it was not until two centuries had passed that in 1872 Karl Weierstrass presented the first definition of a function with a graph that would today be considered fractal, having the non-intuitive property of being everywhere continuous but nowhere differentiable.

The set of regions of a map can be represented more abstractly as an undirected graph that has a vertex for each region and an edge for every pair of regions that share a boundary segment.

This graph is planar ( it is important to note that we are talking about the graphs that have some limitations according to the map they are transformed from only ): it can be drawn in the plane without crossings by placing each vertex at an arbitrarily chosen location within the region to which it corresponds, and by drawing the edges as curves that lead without crossing within each region from the vertex location to each shared boundary point of the region.

Conversely any planar graph can be formed from a map in this way.

In graph-theoretic terminology, the four-color theorem states that the vertices of every planar graph can be colored with at most four colors so that no two adjacent vertices receive the same color, or for short, " every planar graph is four-colorable " (; ).

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.

This causes redundancy in an undirected graph: for example, if vertices A and B are adjacent, A's adjacency list contains B, while B's list contains A. Adjacency queries are faster, at the cost of extra storage space.

It is a directed or undirected graph consisting of vertices, which represent concepts, and edges.

TSP can be modelled as an undirected weighted graph, such that cities are the graph's vertices, paths are the graph's edges, and a path's distance is the edge's length.

In the symmetric TSP, the distance between two cities is the same in each opposite direction, forming an undirected graph.

Given a connected, undirected graph, a spanning tree of that graph is a subgraph that is a tree and connects all the vertices together.

More generally, any undirected graph ( not necessarily connected ) has a minimum spanning forest, which is a union of minimum spanning trees for its connected components.

A path in an undirected graph is a sequence of vertices

In mathematics, more specifically graph theory, a tree is an undirected graph in which any two vertices are connected by exactly one simple path.

A tree is an undirected simple graph G that satisfies any of the following equivalent conditions:

A forest is an undirected graph, all of whose connected components are trees ; in other words, the graph consists of a disjoint union of trees.

Equivalently, a forest is an undirected cycle-free graph.

A polytree or oriented tree is a directed graph with at most one undirected path between any two vertices.

In other words, a polytree is a directed acyclic graph for which there are no undirected cycles either.

Cayley's formula is the special case of complete graphs in a more general problem of counting spanning trees in an undirected graph, which is addressed by the matrix tree theorem.

In computer science, Prim's algorithm is a greedy algorithm that finds a minimum spanning tree for a connected weighted undirected graph.

* In graph theory, a complete graph is an undirected graph in which every pair of vertices has exactly one edge connecting them.