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 connected graph with only countably many vertices admits a normal spanning tree.

Every aspect of life, every word, plant, animal and ritual was connected to the power and authority of the gods.

Every device connected to one of its ports can send packets to any of the others.

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 vertex of an-dimensional box is connected to edges.

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 invariant and co-variant thus becomes expressible by a graph precisely identical with a Kekuléan diagram or chemicograph.

* Every tree is a bipartite graph and a median graph.

Every tree with only countably many vertices is a planar graph.

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 planar graph whose faces all have even length is bipartite.

Every vertex of this graph has an even degree, therefore this is an Eulerian graph.

Every outerplanar graph is a planar graph.

Every outerplanar graph is also a subgraph of a series-parallel graph.

Every forest, and every cactus graph is outerplanar.

Every maximal outerplanar graph with n vertices has exactly 2n − 3 edges, and every bounded face of a maximal outerplanar graph is a triangle.

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 group G acts on G, i. e. in two natural but essentially different ways:, or.

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 〉 extends an equivalence of certain subcategories.

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 ).

Water for Every Farm: A practical irrigation plan for every Australian property, K. 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 reduced word is an alternating product of elements of G and elements of H, e. g.

* 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.