Graphs with weights, or weighted graphs, are used to represent structures in which pairwise connections have some numerical values.

Graphs whose edges or vertices have names or labels are known as labeled, those without as unlabeled.

Graphs that admit exact colorings have been classified.

Graphs may be presented, for example, of coincidence rate against the difference between the settings a and b, but if a more comprehensive set of experiments had been done it might have become clear that the rate depended on a and b separately.

Graphs are represented graphically by drawing a dot or circle for every vertex, and drawing an arc between two vertices if they are connected by an edge.

Graphs with labeled vertices only are vertex-labeled, those with labeled edges only are edge-labeled.

Graphs are frequently drawn as node-link diagrams in which the vertices are represented as disks or boxes and the edges are represented as line segments, polylines, or curves in the Euclidean plane.

Graphs are an expressive, visual and mathematically precise formalism for modelling of objects ( entities ) linked by relations ; objects are represented by nodes and relations between them by edges.

* Graphs are used to visualize relationships between different quantities.

Graphs are basic objects in combinatorics.

Graph ( mathematics ) | Graphs like this are among the objects studied by discrete mathematics, for their interesting graph property | mathematical properties, their usefulness as models of real-world problems, and their importance in developing computer algorithm s.

Graphs are one of the prime objects of study in discrete mathematics.

Graphs are among the most ubiquitous models of both natural and human-made structures.

Graphs are one of the objects of study in discrete mathematics.

Graphs are the basic subject studied by graph theory.

( Graphs which show just the length of the rule in the two dynasties are the most widely known ; however, Fomenko's conclusions are also based on other parameters, as described above.

Graphs are converted into factor graph form to perform belief propagation.

Current research activities of the Department are in the areas of Analysis, Algebra, Operator Theory, Functional Analysis, General topology, Fuzzy mathematics, Graph Theory, Combinations, Convexity Theory, Fluid Dynamics, Non-linear waves, Stability, Stochastic Processes in general and Random Graphs, Operations Research and the History of Mathematics.

Graphs showing this trade-off are available from folds. net.

The optional modules are Number Patterns, Geometry and Trigonometry, Graphs and Relations, Business-Related Mathematics, Networks and Decision Mathematics, or Matrices.

40 Graphs, w species pictures, also Tables, Photos, etc.

Mikhalev, Monoids, Acts and Categories: with Applications to Wreath Products and Graphs, De Gruyter Expositions in Mathematics vol.

A Discussion of the Two qi 其 Graphs in the First Chapter of the Daodejing .” PEW 60. 3 ( 2010 ): 391-421

Free groups of higher rank: Graphs or punctured plane.

* Chein, M., Mugnier, M .- L. ( 2009 ), Graph-based Knowledge Representation: Computational Foundations of Conceptual Graphs, Springer, 2009, ISBN 978-1-84800-285-2.

It is also possible to represent logical descriptions using semantic networks such as the existential Graphs of Charles Sanders Peirce or the related Conceptual Graphs of John F. Sowa.

Stirling Numbers of the First Kind., § 24. 1. 3 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.

* Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions ( with Formulas, Graphs and Mathematical Tables ), U. S. Dept.

Coxeter, R. Frucht and D. L. Powers, Zero-Symmetric Graphs, ( 1981 ) Academic Press.

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 self-dual polyhedron must have the same number of vertices as faces.

For four or more points on the same circle ( e. g., the vertices of a rectangle ) the Delaunay triangulation is not unique: each of the two possible triangulations that split the quadrangle into two triangles satisfies the " Delaunay condition ", i. e., the requirement that the circumcircles of all triangles have empty interiors.

Since the proving of the theorem, efficient algorithms have been found for 4-coloring maps requiring only O ( n < sup > 2 </ sup >) time, where n is the number of vertices.

First, if planar regions separated by the graph are not triangulated, i. e. do not have exactly three edges in their boundaries, we can add edges without introducing new vertices in order to make every region triangular, including the unbounded outer region.

Since charge is preserved, some vertices still have positive charge.

A cubic graph ( all vertices have degree three ) of girth – that is as small as possible – is known as a-cage ( or as a ( 3 ,)- cage ).

Let a trapezoid have vertices A, B, C, and D in sequence and have parallel sides AB and CD.

( If both planes have the same number of vertices,

We also have a special vertex or vertices representing the local variables and references held by the runtime system, and no edges ever go to these nodes, although edges can go from them to other nodes.

After the appropriate calculations have been performed to transform the 3D coordinates of the vertices into 2D screen coordinates, a naïve interpretation could create a wireframe representation by simply drawing straight lines between the screen coordinates of the appropriate vertices using the edge list.

* For any three vertices in a tree, the three paths between them have exactly one vertex in common.

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.

When the number of vertices in the graph is known ahead of time, and additional data structures are used to determine which vertices have already been added to the queue, the space complexity can be expressed as where is the cardinality of the set of vertices.

A corollary of the law of sines as stated above is that in a tetrahedron with vertices O, A, B, C, we have

In graphs in which all vertices have odd degree, an argument related to the handshaking lemma shows that the number of Hamiltonian cycles through any fixed edge is always even, so if one Hamiltonian cycle is given, then a second one must also exist.