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

* 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 topological space X is a dense subspace of a compact space having at most one point more than X, by the Alexandroff one-point compactification.

* Every finite topological space gives rise to a preorder on its points, in which x ≤ y if and only if x belongs to every neighborhood of y, and every finite preorder can be formed as the specialization preorder of a topological space in this way.

* Every topological group is completely regular.

Every group can be trivially made into a topological group by considering it with the discrete topology ; such groups are called discrete groups.

Every topological group can be viewed as a uniform space in two ways ; the left uniformity turns all left multiplications into uniformly continuous maps while the right uniformity turns all right multiplications into uniformly continuous maps.

Every subgroup of a topological group is itself a topological group when given the subspace topology.

Every topological ring is a topological group ( with respect to addition ) and hence a uniform space in a natural manner.

Every local field is isomorphic ( as a topological field ) to one of the following:

* Every non-empty Baire space is of second category in itself, and every intersection of countably many dense open subsets of X is non-empty, but the converse of neither of these is true, as is shown by the topological disjoint sum of the rationals and the unit interval 1.

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 finite-dimensional Hausdorff topological vector space is reflexive, because J is bijective by linear algebra, and because there is a unique Hausdorff vector space topology on a finite dimensional vector space.

Every Boolean algebra can be obtained in this way from a suitable topological space: see Stone's representation theorem for Boolean algebras.

Every such regular cover is a principal G-bundle, where G = Aut ( p ) is considered as a discrete topological group.

Every Boolean algebra is a Heyting algebra when a → b is defined as usual as ¬ a ∨ b, as is every complete distributive lattice when a → b is taken to be the supremum of the set of all c for which a ∧ c ≤ b. The open sets of a topological space form a complete distributive lattice and hence a Heyting algebra.

* Every constant function between topological spaces is continuous.

Every topological group is an H-space ; however, in the general case, as compared to a topological group, H-spaces may lack associativity and inverses.

Every interior algebra can be represented as a topological field of sets with its interior and closure operators corresponding to those of the topological space.

Every separable topological space is ccc.

Every metric space which is ccc is also separable, but in general a ccc topological space need not be separable.

Every locally compact group which is second-countable is metrizable as a topological group ( i. e. can be given a left-invariant metric compatible with the topology ) and complete.