Graph (mathematics) (5614 bytes)
1: ...s the basic definitions. For a broader view see [[graph theory]].''
3: :''For another mathematical use of "graph", see [[graph of a function]].''
5: [[Image:6n-graf.png|right|frame|A graph with 6 vertices and 7 edges.]]
7: ...nnected by links called '''edges'''. Typically, a graph is depicted as a set of dots (the vertices) conne...
11: Definitions in graph theory vary in the literature. Here are the conve...

Graph (836 bytes)
1: A '''graph''' is
3: * In [[mathematics]], the word '''graph''' can have three meanings:
4: ...* In [[graph theory]], a '''[[graph (mathematics)|graph]]''' is an abstract object consisting of [[vertex...
5: ** The [[graph of a function]] ''f'' : ''X'' -> ''Y'' is the set...
6: ... [[graph of a relation]], a generalisation of the graph of a function.

Graph reduction (2294 bytes)
1: ...quivalent [[sub-expression]]s of an [[Expression (mathematics)|expression]]. When [[evaluation|evaluating]] the...
35: ...also be represented as a [[graph (data structure)|graph]], allowing sub-expressions to be shared:
46: ...duction also applies to graphs. Hence we have '''graph reduction'''.
48: Now evaluation with outermost graph reduction can proceed as follows:
64: ... referred to as [[lazy evaluation]] and innermost graph reduction is referred to as [[eager evaluation]].

Graph of a function (1907 bytes)
1: ... in the form of a curve, together with axes, etc. Graphing is sometimes referred to as '''curve sketching...
3: The graph of the function
7: The graph of the cubic polynomial on the real line
13: ...ed of mathematical statements, e.g., the [[closed graph theorem]] in [[functional analysis]].
15: ...s with different [[codomain]] could have the same graph. For example, the cubic polynomial mentioned abo...

Regular graph (1148 bytes)
1: ...ory)|valency]] ''k'' is called a '''''k''-regular graph'''.
3: ...2-regular graph consists of disconnected [[cycle (graph theory)|cycle]]s.
5: A 3-regular graph is know as a [[cubic graph]].
7: ...gular are the [[cycle graph]] and the [[circulant graph]] on 6 vertices.
9: The [[complete graph]] <math>K_m</math> is strongly regular for any <m...

Complete graph (1458 bytes)
1: ...rtite graph]] <math>K_{3,3}</math>) as a [[minor (graph theory)|minor]].
3: Complete graphs on <math>n</math> vertices, for <math>n</math> b...
5: [[Image:Complete graph K1.png|thumb|left|200px|''K''₁]]
6: [[Image:Complete graph K2.png|thumb|left|200px|''K''₂]]
7: [[Image:Complete graph K3.png|thumb|left|200px|''K''₃]]

Inverse graph (292 bytes)
1: ...he '''inverse graph''' of a [[graph]], G, is the graph with the exact same [[vertex|vertices]] as G, but...
3: [[Category:Graphs]]

Petersen graph (4731 bytes)
1: ...Petersen graph.png|thumb|200px|right|The Petersen graph]]
2: ...thumb|200px|right|Another drawing of the Petersen graph, with only two crossings]]
3: [[Image:Petersen graph unit-distance.png|thumb|200px|right|Another drawi...
5: ...erves as a useful example and counterexample in [[graph theory]]. It was first published by [[Julius Pet...
10: The Petersen graph ...

Visibility graph (1526 bytes)
1: ...isible locations. Each node or [[vertex]] in the graph represents a point location, and each [[edge]] re...
3: Visibility graphs are classically regarded as two dimensional arte...
5: ...s of a supposed art gallery. For the purposes of mathematics, an art gallery is simply an arbitrary polygon wi...
9: ...hitecture and urban planning through [[visibility graph analysis]].
17: [[Category:Graph theory]]

Graph coloring (7744 bytes)
1: ...e:3-coloringEx.png|framed|A 3-coloring suits this graph, but fewer colors would result in adjacent vertic...
3: ...o certain objects in a [[Glossary of graph theory|graph]].
6: ...a graph is just the vertex coloring of its [[line graph]].
7: ... is just the vertex coloring of its [[Glossary of graph theory|(planar) dual]].
10: Graph coloring is not to be confused with [[graph labeling]], which is an assignment of ''labels'',...

Universal graph (293 bytes)
1: ...finite (or at-most-[[countable]]) graph as a [[subgraph]].
3: ... graph" was sometimes used to denote a [[complete graph]].
6: [[category:Graphs]]

Scene graph (5558 bytes)
1: ...[Adobe Illustrator]] and [[CorelDraw]]. The scene graph contains the pictorial data items being edited an...
3: ...ne nodes]], it is practical to think of the scene graph as composed of shapes rather than going to a lowe...
5: ...are also done via linear searches. For small scenegraphs, this tends to suffice.
7: ...his case, the scenegraph can contain smaller scenegraphs as nodes, and formal type declarations of such s...
9: These "subscenegraphs", depending on the application, can be known to ...

Turan graph (888 bytes)
1: ... ''T''(''n'',''r'') is the complete ''r''-partite graph with ''n'' vertices whose partite sets differ in ...
3: Think about this graph as a process: First you take place for the ''r'' ...
11: ...e (with respect to the number of edges) among all graphs with ''n'' vertices (see [[Turan's theorem]]).
16: * [[Extremal graph theory]]

Critical graph (2517 bytes)
2: ...to the [[Graph coloring|chromatic number]] of a [[graph]].
3: ...atic number, which is a very important measure in graph theory.
6: ...ertex or an edge is a '''critical element''' of a graph ''G'' if its deletion would decrease the chromati...
7: ...viously such decrement can be no more than 1 in a graph.
9: A '''critical graph''' is a graph in which every vertex or edge is a critical eleme...

Cycle graph (1009 bytes)
1: ... of a [[cycle (graph theory)|cycle]]. The circle graph with <math>n</math> vertices is called <math>C_n<...
3: ... [[diconnected (graph theory)|diconnected]] cycle graph, that is all [[directed edge]]s in the cycle poin...
7: * A circle graph is
8: * [[connected graph|connected]]
9: * [[regular graph|2-regular]].

Graph theory (9461 bytes)
1: ...]]s (or Arcs) which can be directed. Typically, a graph is designed as a set of dots (the vertices) conne...
4: <tr><td>''A graph with 6 vertices and 7 edges.''</td></tr></table>
6: ...'B''. The development of [[algorithm]]s to handle graphs is therefore of major interest in [[computer sci...
8: ...igraph with weighted edges is called a [[network (mathematics)|network]].
10: ...network is much looser, and may often be a simple graph.

Expander graph (1652 bytes)
1: ...ns to [[computer science]], in particular [[cryptography]] and the theory of [[error-correcting code]]s.
3: ...h probability. Explicit constructions of expander graphs are needed in several applications.
5: ...he adjacency matrix. They are formally defined as graphs whose second eigenvalue is less than
11: The ''vertex expansion'' of a graph is the minimum of |'''N'''('''S''')| / |'''S'''|.
15: The ''edge expansion'' of a graph is the minimum ''E''('''S''',/'''S''') / |'''S'''...

Graph homomorphism (2617 bytes)
1: ...'graph homomorphism''' is a mapping between two [[graphs]] that respects their structure. More concretely...
9: ...>f</math> from a graph <math>G:=(V,E)</math> to a graph <math>G':=(V',E')</math> is a function
16: ...ve multiedges and loops. In the case of [[simple graph]]s, the definition can is slightly simpler: where...
18: ...wed to map to a vertex. Thus, if the destination graph has no loops, adjacent vertices can't map to the ...
20: ...he two graphs are identical from the viewpoint of graph theory.

Cubic graph (363 bytes)
1: ...3. In other words a cubic graph is a 3-[[regular graph]].
4: * [[Petersen graph]]
5: * [[snark (graph theory)|Snarks]]
9: [[Category:Graphs]]

Perfect graph (3227 bytes)
1: [[Category:Graphs]][[Category:Theorems]]
2: ...of graph theory#Clique|clique number]] of that subgraph.
4: Some of the more well-known perfect graphs are
5: * the [[line graph]] of a [[bipartite graph]]
6: * ''interval graphs'' (vertices represent line intervals; and edges,...

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