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{{Short description|Topological space arising from a usual graph}} ...nd <math>1</math> with the point associated to <math>y</math>. That is, as topological spaces, graphs are exactly the [[simplicial complex|simplicial 1-complex]]e ...

[[File:Topological Rose.png|thumb|upright=0.8|<math>B_4</math>, a bouquet with one vertex and ...ace of <math>m</math> circles joined at a point. When the context of graph theory is clear, it can be called more simply a '''bouquet'''.{{r|bw}} ...

[[File:Uniform tree.svg|thumb|300px|Section of the uniform tree for graph <math>G</math>]] ...[[automorphism group]] G=Aut(X) of the tree, which is a [[locally compact topological group]], is [[unimodular group|unimodular]] and G\X is finite. Also equival ...

{{short description|Partially ordered topological space}} ...h a closed [[partial order]] <math>\leq</math>, i.e. a partial order whose graph <math>\{(x, y) \in X^2 \mid x \leq y\}</math> is a closed subset of <math>X ...

{{Short description|Graph describing a topological embedding}} ...mb|upright=1.5|A graph-encoded map (gray triangles and colored edges) of a graph in the plane (white circles and black edges)]] ...

...edges (two of them twisted), and one face. It represents an embedding of a graph with three self-loops onto the connected sum of three [[projective plane]] ...raph embedding]]s, equivalent in power to signed [[rotation system]]s or [[graph-encoded map]]s.{{r|d}} It is convenient for visualizations of embeddings, b ...

...e application of [[topology|topological]] and [[algebraic topology|algebro-topological]] methods to solving problems in [[combinatorics]]. ...the [[Kneser graph|Kneser conjecture]], thus beginning the new field of '''topological combinatorics'''. Lovász's proof used the [[Borsuk–Ulam theorem]] and this ...

{{Short description|Topological index of a molecule used in biochemistry}} In [[chemical graph theory]], the '''Szeged index''' is a [[topological index]] of a [[molecule]], used in [[biochemistry]]. The Szeged index, int ...

{{distinguish|text = the [[Heawood graph]], which is a member of the Heawood graph family}} ...erm '''Heawood family''' refers to either one of the following two related graph families generated via [[ΔY- and YΔ-transformations]]: ...

...>f</math> and the identity <math>1_X</math>. The diagonal embedding is the graph morphism of <math>1_X</math>. Notice that a [[topological space]] ''Y'' is [[Hausdorff topological space|Hausdorff]] iff the diagonal embedding ...

{{Short description|Topological graph theory}} ...tended to all cellular graph embeddings (embeddings all of whose faces are topological disks) by {{harvtxt|Lins|1982}}.{{r|l}} ...

...theory]], the '''hyper-Wiener index''' or '''hyper-Wiener number''' is a [[topological index]] of a [[molecule]], used in [[biochemistry]]. The hyper-Wiener index ...], introduced by [[Harry Wiener]]. The hyper-Wiener index of a [[connected graph]] ''G'' is defined by ...

...e red component) has an odd number of edges, the minimum possible for this graph.]] ...maining graph <math>G-T</math>, the number of [[Connected component (graph theory)|connected components]] with an odd number of edges is as small as possible ...

...act. The amalgamation can then be used to study properties of the original graph in an easier to understand context. Applications include embeddings,<ref na Note that while <math>G</math> can be a graph or a [[pseudograph]], it will usually be the case that <math>H</math> is a ...

...elled '''infra[[Wiktionary:barreled|barreled]]''') if every [[Bounded set (topological vector space)|bounded]] [[Barreled space|barrel]] is a neighborhood of the Similarly, '''quasibarrelled spaces''' are [[topological vector spaces]] (TVS) for which every [[bornivorous]] barrelled set in the ...

...cessarily [[Continuous map|continuous]]. A general version of the [[closed graph theorem]] holds for ultrabornological spaces. Let <math>X</math> be a [[topological vector space]] (TVS). ...

...group|generating set]] of ''G'', and a [[subgroup]] of''G''. The Schreier graph encodes the abstract structure of the group modulo an [[equivalence relatio The graph is named after [[Otto Schreier]], who used the term "{{lang|de|Nebengruppen ...

...heorems in [[functional analysis]]|other results with the same name|Closed graph theorem}} ...such an operator is called a [[closed linear operator]]; see also [[closed graph property]]). ...

In [[chemical graph theory]], the '''Padmakar–Ivan (PI) index''' is a [[topological index]] of a [[molecule]], used in [[biochemistry]]. The Padmakar–Ivan inde ...ener index]], introduced by [[Harry Wiener]]. The Padmakar–Ivan index of a graph ''G'' is the sum over all edges ''uv'' of ''G'' of number of edges which ...

{{Short description|Element of graph theory}} ...each cycle edge that make the resulting directed graph [[Directed acyclic graph|acyclic]]. Two orientations are shown as adjacent when they differ in the d ...