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#REDIRECT [[Dijkstra's algorithm#Practical optimizations and infinite graphs]] ...

#REDIRECT [[Dijkstra's algorithm#Practical optimizations and infinite graphs]] ...

...ath]]s with one extra edge attached to the vertex next to an endpoint; the infinite families with spectral radius exactly 2 are the [[Cycle (graph theory)|cycl | title = Strongly regular graphs having strongly regular subconstituents ...

...example, the edges of the [[truncated tetrahedron]]), and not all regular graphs are vertex-transitive (for example, the [[Frucht graph]] and [[Tietze's gra ...phs on at most 1280 vertices.<ref>{{citation|title=Cubic vertex-transitive graphs on up to 1280 vertices|author1=Potočnik P., Spiga P. |author2=Verret G. |n ...

...labeled vertices.<ref>{{Cite OEIS|A061540|name=Number of connected labeled graphs with n nodes and n+1 edges}}</ref> ...

In [[mathematics]], a '''universal graph''' is an infinite [[Graph (discrete mathematics)|graph]] that contains ''every'' finite (or | title = Universal graphs and universal functions ...

==Directed graphs== ...these statements in both edge and vertex versions remain true in directed graphs (when considering directed paths). ...

{{unsolved|mathematics|Are graphs uniquely determined by their subgraphs?}} Informally, the '''reconstruction conjecture''' in [[graph theory]] says that graphs are determined uniquely by their subgraphs. It is due to [[Paul Kelly (mat ...

...rtite graph]]s ''K_m,n'' where ''m'' = ''n'', the density of the graphs approaches 1/2 as the number of vertices approaches infinity, and so the fa The '''graph density''' of simple graphs is defined to be the ratio of the number of edges {{math|{{abs|''E''}}}} wi ...

{{short description|Characterization of graphs with perfect matchings}} ...is a characterization of finite undirected [[Graph (discrete mathematics)|graphs]] with [[perfect matching]]s. It is a special case of the [[Tutte–Berge for ...

{{short description|Characterization of graphs with perfect matchings}} ...is a characterization of finite undirected [[Graph (discrete mathematics)|graphs]] with [[perfect matching]]s. It is a special case of the [[Tutte–Berge for ...

...umber=A007721|name=Number of distinct degree sequences among all connected graphs with n nodes}}</ref> and 68 [[matroid]]s on four labeled elements.<ref>{{Ci ...g square.<ref>{{Cite OEIS|sequencenumber=A018842|name=Number of squares on infinite chess-board at ''n'' knight's moves from center}}</ref> ...

{{For|the family of graphs known as paths|Path graph}} ...ipath'''{{sfn|McCuaig|1992|p=205}}) in a [[directed graph]] is a finite or infinite sequence of edges which joins a sequence of distinct vertices, but with the ...

...graphs between selected vertices. Some have no defined distance (marked as infinite distance) because they are in different [[component (graph_theory)|connecte |title = Geodesic distance in planar graphs ...

...orth method''' is a method for showing [[isomorphism]] between [[countably infinite]] structures satisfying specified conditions. In particular it can be used * any two [[countably infinite]] [[dense order|densely ordered]] sets (i.e., linearly ordered in such a wa ...

...t map to {{mvar|c—d}}, but not to {{mvar|d—c}}. [[Star (graph theory)|Star graphs]] are a simple example of being edge-transitive without being vertex-transi ...n an arc-transitive graph. Such a definition would include half-transitive graphs, which are excluded under the definition above. ...

...that [[Wagner's theorem]] characterizes the [[planar graph]]s as being the graphs that do not have the [[complete graph]] <math>K_5</math> or the [[complete ...nonetheless be considered as distinct objects, then the minor ordering on graphs forms a [[preorder]], a relation that is reflexive and transitive but not n ...

...st3 = M. A. | doi = 10.1038/nature03204 | title = Evolutionary dynamics on graphs | journal = Nature | volume = 433 | issue = 7023 | pages = 312–316 | year = One property studied on graphs with two types of individuals is the ''fixation probability'', which is def ...

[[File:Infinite lattice of divisors.svg|thumb|[[Hasse diagram]] of the [[natural number]]s, == Comparability graphs == ...

...rdős]]; it states that the Ramsey number of graphs in any sparse family of graphs should [[linear growth|grow linearly]] in the number of [[vertex (graph the ...{harvtxt|Rödl|Thomas|1991}}.</ref> It is also known for subdivided graphs, graphs in which no two adjacent vertices have degree greater than two.<ref>{{harvt ...