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* Every connected graph G admits a spanning tree, which is a tree that contains every vertex of G and whose edges are edges of G.

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## Some Related Sentences

Every and connected

__Every__

__connected__

**graph**

**is**an expander ; however

**,**different

__connected__graphs have different expansion parameters

**.**

__Every__individual

**is**

__connected__with the rest

**of**the world

**,**

**and**the universe

**is**fashioned for universal harmony

**.**

__Every__aspect

**of**life

**,**

**every**word

**,**plant

**,**animal

**and**ritual was

__connected__to the power

**and**authority

**of**the gods

**.**

__Every__aspect

**of**life

**,**

**every**word

**,**plant

**,**animal

**and**ritual was

__connected__to the power

**and**authority

**of**the gods

**.**

The generalized Poincaré conjecture states

**that**__Every__simply__connected__**,**closed n-manifold**is**homeomorphic to the n-sphere**.**__Every__closed 3-manifold has

**a**prime decomposition: this means it

**is**the

__connected__sum

**of**prime three-manifolds ( this decomposition

**is**essentially unique except for

**a**small problem in the case

**of**non-orientable manifolds ).

__Every__part

**of**it

**,**including the blue

**and**white colors ( see below ), the cross

**,**as well as the stripe arrangement can be

__connected__to very old historical elements ; however it

**is**difficult to establish " continuity ", especially as there

**is**no record

**of**the exact reasoning behind its official adoption in early 1822

**.**

__Every__Eulerian orientation

**of**

**a**

__connected__

**graph**

**is**

**a**strong orientation

**,**an orientation

**that**makes the resulting directed

**graph**strongly

__connected__

**.**

__Every__maximal outerplanar

**graph**satisfies

**a**stronger condition than Hamiltonicity: it

**is**node pancyclic

**,**meaning

**that**for

**every**

**vertex**v

**and**

**every**k in the range from three to the number

**of**vertices in the

**graph**

**,**there

**is**

**a**length-k cycle containing v

**.**A cycle

**of**this length may be found by repeatedly removing

**a**triangle

**that**

**is**

__connected__to the rest

**of**the

**graph**by

**a**single edge

**,**such

**that**the removed

**vertex**

**is**not v

**,**until the outer face

**of**the remaining

**graph**has length k

**.**

__Every__broadcasting company has members

**and**the number

**of**members gives them

**a**status

**that**

**is**

__connected__to the number

**of**hours

**of**broadcasting

**.**

__Every__

**graph**(

**that**

**is**

__connected__

**and**not

**a**

**tree**) has multiple

**spanning**trees

**,**so we once again have an example where the problem itself allows multiple possible outcomes

**,**

**and**the algorithm chosen can arrive at any one

**of**them

**,**but will never arrive at something else

**.**

This flat may be identified with the partition

**of**the vertices**of**into the__connected__components**of**the subgraph formed by:__Every__set**of****edges**having the same closure as gives rise to the same partition**of**the vertices**,****and**may be recovered from the partition**of**the vertices**,**as it consists**of**the**edges****whose**endpoints both belong to the same set in the partition**.**__Every__rational variety

**,**including the projective spaces

**,**

**is**rationally

__connected__

**,**but the converse

**is**false

**.**

*****

__Every__external device

__connected__to the Freebox player

**is**available to

**that**device only but the devices

__connected__to the Freebox Server

**are**available to

**every**Freebox player

__connected__

**.**

__Every__extra foot

**of**cord increases the electrical resistance

**,**

**which**decreases the power the cord can deliver to

__connected__devices

**.**

__Every__closed

**,**orientable

**,**

__connected__3-manifold

**is**obtained by performing Dehn surgery on

**a**link in the 3-sphere

**.**

__Every__

__connected__symmetric

**graph**must thus be both vertex-transitive

**and**edge-transitive

**,**

**and**the converse

**is**true for graphs

**of**odd degree

**.**

Every and graph

:"[...]

__Every__invariant**and**co-variant thus becomes expressible by**a**__graph__precisely identical with**a**Kekuléan diagram or chemicograph**.**
In mathematics

**,**Tait's conjecture states**that**"__Every__3-connected planar cubic__graph__has**a**Hamiltonian cycle ( along the**edges**) through all its vertices ".__Every__homomorphism

**of**the Petersen

__graph__to itself

**that**doesn't identify adjacent vertices

**is**an automorphism

**.**

__Every__directed acyclic

__graph__has

**a**topological ordering

**,**an ordering

**of**the vertices such

**that**the starting endpoint

**of**

**every**edge occurs earlier in the ordering than the ending endpoint

**of**the edge

**.**

__Every__maximal outerplanar

__graph__with n vertices has exactly 2n − 3

**edges**

**,**

**and**

**every**bounded face

**of**

**a**maximal outerplanar

__graph__

**is**

**a**triangle

**.**

Every and G

Group actions / representations:

__Every__group__G__can be considered as**a**category with**a**single object**whose**morphisms**are**the elements**of**__G__**.**A functor from__G__to Set**is**then nothing but**a**group action**of**__G__on**a**particular set**,**i**.**e**.****a**G-set**.**__Every__homomorphism f:

__G__→ H

**of**Lie groups induces

**a**homomorphism between the corresponding Lie algebras

**and**

**.**

__Every__open subgroup H

**is**also closed

**,**since the complement

**of**H

**is**the open set given by the union

**of**open sets gH for g in

__G__

__Every__adjunction 〈 F

**,**

__G__

**,**ε

**,**η 〉 gives rise to an associated monad 〈 T

**,**η

**,**μ 〉 in the category D

**.**The functor

__Every__inner automorphism

**is**indeed an automorphism

**of**the group

__G__

**,**i

**.**e

**.**it

**is**

**a**bijective map from

__G__to

__G__

**and**it

**is**

**a**homomorphism ; meaning ( xy )< sup >

**a**</ sup >

__Every__non-inner automorphism yields

**a**non-trivial element

**of**Out (

__G__), but different non-inner automorphisms may yield the same element

**of**Out (

__G__).

__Every__one

**of**the infinitely many vertices

**of**

__G__can be reached from v < sub > 1 </ sub > with

**a**simple path

**,**

**and**each such path must start with one

**of**the finitely many vertices adjacent to v < sub > 1 </ sub >.

__Every__smooth function

__G__over the symplectic manifold generates

**a**one-parameter family

**of**symplectomorphisms

**and**if

__Every__year at the International Astronautical Congress

**,**three prestigious awards

**are**given out: the Allan D

**.**Emil Memorial Award

**,**the Franck J

**.**Malina Astronautics Medal

**and**the Luigi

__G__

**.**Napolitano Award

**.**

__Every__such regular cover

**is**

**a**principal G-bundle

**,**where

__G__= Aut ( p )

**is**considered as

**a**discrete topological group

**.**

*****

__Every__Polish space

**is**homeomorphic to

**a**

__G__< sub > δ </ sub > subspace

**of**the Hilbert cube

**,**

**and**

**every**

__G__< sub > δ </ sub > subspace

**of**the Hilbert cube

**is**Polish

**.**

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