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| {{Redirect-distinguish|Pseudograph|Pseudepigraph}}
| | == as to enhance the strength == |
| {{About|the mathematical concept}}
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| [[File:Multi-pseudograph.svg|thumb|right|A multigraph with multiple edges (red) and several loops (blue). Not all authors allow multigraphs to have loops.]]
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| In [[mathematics]], and more specifically in [[graph theory]], a '''multigraph''' (or '''pseudograph''') is a [[graph (mathematics)|graph]] which is permitted to have [[multiple edges]] (also called "parallel edges"<ref>For example, see Balakrishnan, p. 1.</ref>), that is, edges that have the same end [[Vertex (graph theory)|nodes]]. Thus two vertices may be connected by more than one edge. (A multigraph is thus different from a [[hypergraph]], which is a graph in which an edge can connect any number of nodes, not just 2.)
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| There are two distinct notions of multiple edges. One says that, as in graphs without multiple edges, the identity of an edge is defined by the nodes it connects, but the same edge can occur several times between these nodes. Alternatively, one defines edges to be first-class entities like nodes, each having its own identity independent of the nodes it connects.
| | ...<br><br>truly boundless existence, the universe [http://www.nrcil.net/fancybox/lib/rakuten_LV_43.html ルイヴィトン バッグ 人気] is filled with an endless sea of chaotic flow, giving birth to many dangerous places, which is the most dangerous of the three Jedi - dump peak circles, stream mountains, the universe boat.<br><br>addition to some dangerous place, the universe is the most common sea [http://www.nrcil.net/fancybox/lib/rakuten_LV_25.html ルイヴィトン2014新作] dolphin universe!<br><br>universe divided into two categories.<br><br>a class of natural sea but the [http://www.nrcil.net/fancybox/lib/rakuten_LV_23.html ルイヴィトン タイガ] universe born out of the 'original universe', the original universe is the universe's most powerful presence in the sea, the [http://www.nrcil.net/fancybox/lib/rakuten_LV_66.html ルイヴィトンデニムバッグ] original universe was born, bred Evolution millions of ethnic groups, multiply the peak, decline, shattered. This is a reincarnation! After the big burst return to the origin, then [http://www.nrcil.net/fancybox/lib/rakuten_LV_11.html ルイヴィトン 靴] a new [http://www.nrcil.net/fancybox/lib/rakuten_LV_79.html 中古ルイヴィトン] cycle will begin again ...... original universe, then the birth of hundreds of millions of such groups, the birth of numerous strong.<br><br>II is the strongest of the universe created the universe.<br><br>strong practice of many layers.<br><br>immortal, they have the [http://www.nrcil.net/fancybox/lib/rakuten_LV_106.html ルイヴィトン バッグ モノグラム] kingdom of God.<br><br>as to enhance the strength, the gradual improvement of the kingdom of God, [http://www.nrcil.net/fancybox/lib/rakuten_LV_50.html ルイヴィトン デザイナー] become the master of the universe |
| | | 相关的主题文章： |
| ==Undirected multigraph (edges without own identity)==
| | <ul> |
| Formally, a multigraph ''G'' is an [[ordered pair]] ''G'':=(''V'', ''E'') with
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| *''V'' a [[Set (mathematics)|set]] of ''vertices'' or ''nodes'',
| | <li>[http://gomusic.mobi/elgg/blog/view/393660/luo-feng-walked-beside-the-speed-tester-to-see http://gomusic.mobi/elgg/blog/view/393660/luo-feng-walked-beside-the-speed-tester-to-see]</li> |
| *''E'' a [[multiset]] of unordered pairs of vertices, called ''edges'' or ''lines''.
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| | | <li>[http://www.0772sn.com/thread-244033-1-1.html http://www.0772sn.com/thread-244033-1-1.html]</li> |
| Multigraphs might be used to model the possible flight connections offered by an airline. In this case the multigraph would be a [[directed graph]] with pairs of directed parallel edges connecting cities to show that it is possible to fly both ''to'' and ''from'' these locations.
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| | | <li>[http://www.bailufangzhou.com/forum.php?mod=viewthread&tid=243224 http://www.bailufangzhou.com/forum.php?mod=viewthread&tid=243224]</li> |
| Some authors also allow multigraphs to have [[loop (graph theory)|loops]], that is, an edge that connects a vertex to itself,<ref>For example, see. Bollobás, p. 7 and Diestel, p. 25.</ref> while others call these '''pseudographs''', reserving the term multigraph for the case with no loops.<ref>''Graphs, Colourings and the Four-Colour Theorem'', by Robert A. Wilson, 2002, ISBN 0-19-851062-4, [http://books.google.com/books?id=iq0sSnIxJioC&pg=PA6&dq=pseudograph&lr=&ei=R-jrSKWoCJGgswOv0eiXBw&sig=ACfU3U20xuoH7jZDq-XGqSnfsmC0oE8KjQ p. 6]</ref>
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| | | </ul> |
| ==Directed multigraph (edges without own identity)==
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| A '''multidigraph''' is a directed graph which is permitted to have ''multiple arcs,'' i.e., arcs with the same source and target nodes. A multidigraph ''G'' is an ordered pair ''G'':=(''V'',''A'') with
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| *''V'' a set of ''vertices'' or ''nodes'',
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| *''A'' a multiset of ordered pairs of vertices called ''directed edges'', ''arcs'' or ''arrows''.
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| A '''mixed multigraph''' ''G'':=(''V'',''E'', ''A'') may be defined in the same way as a [[mixed graph]].
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| ==Directed multigraph (edges with own identity)==
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| A multidigraph or [[Quiver (mathematics)|quiver]] ''G'' is an ordered [[tuple|4-tuple]] ''G'':=(''V'', ''A'', ''s'', ''t'') with
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| *''V'' a [[Set (mathematics)|set]] of ''vertices'' or ''nodes'',
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| *''A'' a [[Set (mathematics)|set]] of ''edges'' or ''lines'',
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| *<math>s : A \rightarrow V</math>, assigning to each edge its source node,
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| *<math>t : A \rightarrow V</math>, assigning to each edge its target node.
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| In [[category theory]] a small [[category (mathematics)|category]] can be defined as a multidigraph (with edges having their own identity) equipped with an associative composition law and a distinguished self-loop at each vertex serving as the left and right identity for composition. For this reason, in category theory the term ''graph'' is standardly taken to mean "multidigraph", and the underlying multidigraph of a category is called its '''underlying digraph'''.
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| ==Labeling==
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| Multigraphs and multidigraphs also support the notion of [[graph labeling]], in a similar way. However there is no unity in terminology in this case.
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| The definitions of '''labeled multigraphs''' and '''labeled multidigraphs''' are similar, and we define only the latter ones here.
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| ''Definition 1'': A labeled multidigraph is a [[labeled graph]] with ''labeled'' arcs.
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| Formally: A labeled multidigraph G is a multigraph with ''labeled'' vertices and arcs. Formally it is an 8-tuple <math>G=(\Sigma_V, \Sigma_A, V, A, s, t, \ell_V, \ell_A)</math> where
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| *V is a set of vertices and A is a set of arcs.
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| *<math>\Sigma_V</math> and <math>\Sigma_A</math> are finite alphabets of the available vertex and arc labels,
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| *<math>s\colon A\rightarrow\ V</math> and <math>t\colon A\rightarrow\ V</math> are two maps indicating the ''source'' and ''target'' vertex of an arc,
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| *<math>\ell_V\colon V\rightarrow\Sigma_V</math> and <math>\ell_A\colon A\rightarrow\Sigma_A</math> are two maps describing the labeling of the vertices and arcs.
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| ''Definition 2'': A labeled multidigraph is a [[labeled graph]] with multiple ''labeled'' arcs, i.e. arcs with the same end vertices and the same arc label (note that this notion of a labeled graph is different from the notion given by the article [[graph labeling]]).
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| ==See also==
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| * [[Glossary of graph theory]]
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| * [[Graph theory]]
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| * Balakrishnan, V. K.; ''Graph Theory'', McGraw-Hill; 1 edition (February 1, 1997). ISBN 0-07-005489-4.
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| * Bollobás, Béla; ''Modern Graph Theory'', Springer; 1st edition (August 12, 2002). ISBN 0-387-98488-7.
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| * Diestel, Reinhard; ''Graph Theory'', Springer; 2nd edition (February 18, 2000). ISBN 0-387-98976-5.
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| * Gross, Jonathan L, and Yellen, Jay; ''Graph Theory and Its Applications'', CRC Press (December 30, 1998). ISBN 0-8493-3982-0.
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| * Gross, Jonathan L, and Yellen, Jay; (eds); ''Handbook of Graph Theory''. CRC (December 29, 2003). ISBN 1-58488-090-2.
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| * Harary, Frank; ''Graph Theory'', Addison Wesley Publishing Company (January 1995). ISBN 0-201-41033-8.
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| * Zwillinger, Daniel; ''CRC Standard Mathematical Tables and Formulae'', Chapman & Hall/CRC; 31st edition (November 27, 2002). ISBN 1-58488-291-3.
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| * {{cite journal | first1=Svante | last1=Janson | authorlink1=Svante Janson | first2=Donald E. |last2=Knuth | authorlink2=Donald Knuth| first3=Tomasz | last3=Luczak | first4=Boris | last4=Pittel | title=The birth of the giant component | journal=Random Structures and Algorithms | volume=4 | year=1993 | issue=3 | pages=231–358 | isbn=3-240-04030-3 {{Please check ISBN|reason=Check digit (3) does not correspond to calculated figure.}} | mr=1220220 | doi=10.1002/rsa.3240040303 }}
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| ==External links== | |
| * {{DADS|Multigraph|multigraph}}
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| [[Category:Extensions and generalizations of graphs]]
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| [[de:Graph (Graphentheorie)#Multigraph]]
| |
as to enhance the strength
...
truly boundless existence, the universe ルイヴィトン バッグ 人気 is filled with an endless sea of chaotic flow, giving birth to many dangerous places, which is the most dangerous of the three Jedi - dump peak circles, stream mountains, the universe boat.
addition to some dangerous place, the universe is the most common sea ルイヴィトン2014新作 dolphin universe!
universe divided into two categories.
a class of natural sea but the ルイヴィトン タイガ universe born out of the 'original universe', the original universe is the universe's most powerful presence in the sea, the ルイヴィトンデニムバッグ original universe was born, bred Evolution millions of ethnic groups, multiply the peak, decline, shattered. This is a reincarnation! After the big burst return to the origin, then ルイヴィトン 靴 a new 中古ルイヴィトン cycle will begin again ...... original universe, then the birth of hundreds of millions of such groups, the birth of numerous strong.
II is the strongest of the universe created the universe.
strong practice of many layers.
immortal, they have the ルイヴィトン バッグ モノグラム kingdom of God.
as to enhance the strength, the gradual improvement of the kingdom of God, ルイヴィトン デザイナー become the master of the universe
相关的主题文章：