Each point with abscissa T on the graph represents an intersection between C and Af.

The roots of this equation are just the ordinates of the intersections of the graph of B with a straight line of unit slope through Af in the b-plane ( the plane of the graph of b ).

Specifically, the conceptual diagram graph 1 identifies only three boxes, two ellipses, and four arrows ( and their five labels ), whereas the picture 1 shows much more pictorial detail, with the scores of implied relationships as implicit in the picture rather than with the nine explicit details in the graph.

The graph of a function with a horizontal, vertical, and oblique asymptote.

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

In collaboration with Shen Lin he devised well-known heuristics for two NP-complete optimization problems: graph partitioning and the travelling salesman problem.

The questions range from counting ( e. g., the number of graphs on n vertices with k edges ) to structural ( e. g., which graphs contain Hamiltonian cycles ) to algebraic questions ( e. g., given a graph G and two numbers x and y, does the Tutte polynomial T < sub > G </ sub >( x, y ) have a combinatorial interpretation ?).

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 ".

The derivative of y with respect to x at a is, geometrically, the slope of the tangent line to the graph of f at a.

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.

An ( N, M, D, K, e )- disperser is a bipartite graph with N vertices on the left side, each with degree D, and M vertices on the right side, such that every subset of K vertices on the left side is connected to more than ( 1 − e ) M vertices on the right.

:: A graph database is a kind of NoSQL database that uses graph structures with nodes, edges, and properties to represent and store information.

For example, they allow representation of a directed graph with trees on the nodes.

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.

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.

Logical formulas are discrete structures, as are proofs, which form finite trees or, more generally, directed acyclic graph structures ( with each inference step combining one or more premise branches to give a single conclusion ).

The PPF is a table or graph ( as at the right ) showing the different quantity combinations of the two goods producible with a given technology and total factor inputs, which limit feasible total output.

We have shown that the graph of F contains at least one component whose inverse is the entire interval {0,T}, and whose multiplicity is odd.

The odd girth and even girth of a graph are the lengths of a shortest odd cycle and shortest even cycle respectively.

* Bipartite graph, a graph is bipartite if it has no odd cycles

* Perfect graph, ( or a Berge graph ), a graph is perfect if it has no odd hole and no odd antihole.

If the graph is not Eulerian, it must contain vertices of odd degree.

Euler's argument shows that a necessary condition for the walk of the desired form is that the graph be connected and have exactly zero or two nodes of odd degree.

Since the graph corresponding to historical Königsberg has four nodes of odd degree, it cannot have an Eulerian path.

Such a circuit exists if, and only if, the graph is connected, and there are no nodes of odd degree at all.

As a Kneser graph of the form it is an example of an odd graph.

* A graph is bipartite if and only if it does not contain an odd cycle.

According to the strong perfect graph theorem, the perfect graphs have a forbidden graph characterization resembling that of bipartite graphs: a graph is bipartite if and only if it has no odd cycle as a subgraph, and a graph is perfect if and only if it has no odd cycle or its complement as an induced subgraph.

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

In graph theory, the girth of a graph is the length of a shortest cycle contained in the graph.

More precisely, he showed that a random graph on vertices, formed by choosing independently whether to include each edge with probability has, with probability tending to 1 as goes to infinity, at most cycles of length or less, but has no independent set of size Therefore, removing one vertex from each short cycle leaves a smaller graph with girth greater than in which each color class of a coloring must be small and which therefore requires at least colors in any coloring.

These are illustrated above in the Cycle graph ( algebra ) | cycle graph format, along with the 180 ° edge ( blue arrows ) and 120 ° vertex ( reddish arrows ) rotation s that permutation | permute the tetrahedron through the positions.

In mathematics, Tait's conjecture states that " Every 3-connected planar cubic graph has a Hamiltonian cycle ( along the edges ) through all its vertices ".

The condition that the graph be 3-regular is necessary due to polyhedra such as the rhombic dodecahedron, which forms a bipartite graph with six degree-four vertices on one side and eight degree-three vertices on the other side ; because any Hamiltonian cycle would have to alternate between the two sides of the bipartition, but they have unequal numbers of vertices, the rhombic dodecahedron is not Hamiltonian.

In a Hamiltonian cubic planar graph, such an edge coloring is easy to find: use two colors alternately on the cycle, and a third color for all remaining edges.

Alternatively, a 4-coloring of the faces of a Hamiltonian cubic planar graph may be constructed directly, using two colors for the faces inside the cycle and two more colors for the faces outside.

If this fragment is part of a larger graph, then any Hamiltonian cycle through the graph must go in or out of the top vertex ( and either one of the lower ones ).

* Grinberg's theorem, a necessary condition on the existence of a Hamiltonian cycle that can be used to show that a graph forms a counterexample to Tait's conjecture

Animated graph of a the paths of totality of a solar eclipse cycle.

The smallest graphs that are regular but not strongly regular are the cycle graph and the circulant graph on 6 vertices.

A theorem by Nash-Williams says that every < span class =" texhtml " >< var > k </ var ></ span >‑ regular graph on < span class =" texhtml " > 2 < var > k </ var > + 1 </ span > vertices has a Hamiltonian cycle.

The energy lost per cycle is proportional to the area of the hysteresis loop in the BH graph.

In the mathematical field of graph theory the Hamiltonian path problem and the Hamiltonian cycle problem are problems of determining whether a Hamiltonian path or a Hamiltonian cycle exists in a given graph ( whether directed or undirected ).