An ( N, M, D, K, e )- disperser is a bipartite graph with N vertices on the left side, each with degree D, and M vertices on the right side, such that every subset of K vertices on the left side is connected to more than ( 1 − e ) M vertices on the right.

* A graph is planar if it contains as a minor neither the complete bipartite graph ( See the Three-cottage problem ) nor the complete graph.

However, it has been proposed that this particular organization is only observed in higher eukaryotes and that it evolved from a bipartite organization with the transition from anamniotes to amniotes.

While it is true that a bipartite quantum state must be entangled in order for it to produce non-local correlations, there exist entangled states that do not produce such correlations.

In general, a bipartite pure state ρ is entangled if and only if its reduced states are mixed rather than pure.

For bipartite pure states, the von Neumann entropy of reduced states is the unique measure of entanglement in the sense that it is the only function on the family of states that satisfies certain axioms required of an entanglement measure.

Traditional Islam views the world as bipartite, consisting of the House of Islam, that is, where people live under the Islamic law-the Shariah-and the House of War, that is, where the people do not live under Islamic law, which must be proselytized ( see da ' wah ) using whatever resources available, including, in some traditionalist and conservative interpretations, the use of violence, as holy struggle in the path of Allah, to either convert its inhabitants to Islam, or to rule them under the Shariah ( cf.

dhimmi ); since the abolition of the Caliphate, there has been debate about the proper role of divisions of the world in Islam, and whether the traditional bipartite division is sufficient to meet the needs of the ummah ( world community of Muslims ) and world moving in to the future.

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

The condition that the graph be 3-regular is necessary due to polyhedra such as the rhombic dodecahedron, which forms a bipartite graph with six degree-four vertices on one side and eight degree-three vertices on the other side ; because any Hamiltonian cycle would have to alternate between the two sides of the bipartition, but they have unequal numbers of vertices, the rhombic dodecahedron is not Hamiltonian.

* Barnette's conjecture, a still-open refinement of Tait's conjecture stating that every bipartite cubic polyhedral graph is Hamiltonian.

In more formal graph-theoretic terms, the problem asks whether the complete bipartite graph K < sub > 3, 3 </ sub > is planar.

Alternatively, it is possible to show that any bridgeless bipartite planar graph with n vertices and m edges has by combining the Euler formula ( where f is the number of faces of a planar embedding ) with the observation that the number of faces is at most half the number of edges ( because each face has at least four edges and each edge belongs to exactly two faces ).

The utility graph K < sub > 3, 3 </ sub > is, like all other complete bipartite graphs, a well-covered graph, meaning that every maximal independent set has the same size.

If G is not bipartite

If at any step a vertex has ( visited ) neighbours with the same label as itself, then the graph is not bipartite.

If the search ends without such a situation occurring, then the graph is bipartite.

Displayed are parts of the ( disjoint ) sets A and B together with parts of the mappings f and g. If the set A ∪ B, together with the two maps, is interpreted as a directed graph, then this bipartite graph has several connected components.

In some neighborhood in the f-plane of any ordinary point of the graph, the function f is a single-valued, continuous function.

The graph of f has at least one component whose support is the entire interval Aj.

We have shown that the graph of F contains at least one component whose inverse is the entire interval {0,T}, and whose multiplicity is odd.

Figure 2 is a graph of the mean achievement scores of each group.

If the force required to remove the coatings is plotted against film thickness, a graph as illustrated schematically in Fig. 5 may characteristically result.

* In graph theory an automorphism of a graph is a permutation of the nodes that preserves edges and non-edges.

The b value compresses the graph of the function horizontally if greater than 1 and stretches the function horizontally if less than 1, and like a, reflects the function in the y-axis when it is negative.

The graph of R ( x, y ) is changed by standard transformations as follows:

Although the description sitting-on ( graph 1 ) is more abstract than the graphic image of a cat sitting on a mat ( picture 1 ), the delineation of abstract things from concrete things is somewhat ambiguous ; this ambiguity or vagueness is characteristic of abstraction.

Asymptotes convey information about the behavior of curves in the large, and determining the asymptotes of a function is an important step in sketching its graph.

The line x = a is a vertical asymptote of the graph of the function

It is impossible for the graph of a function to intersect a vertical asymptote ( or a vertical line in general ) in more than one point.

While the numerical difference between the decimal and binary interpretations is relatively small for the prefixes kilo and mega, it grows to over 20 % for prefix yotta, illustrated in the linear-log graph ( at right ) of difference versus storage size.

Binary relations are used in many branches of mathematics to model concepts like " is greater than ", " is equal to ", and " divides " in arithmetic, " is congruent to " in geometry, " is adjacent to " in graph theory, " is orthogonal to " in linear algebra and many more.

The sets X and Y are called the domain ( or the set of departure ) and codomain ( or the set of destination ), respectively, of the relation, and G is called its graph.

Some mathematicians, especially in set theory, do not consider the sets and to be part of the relation, and therefore define a binary relation as being a subset of x, that is, just the graph.

* A Gaussian function, a specific kind of function whose graph is a bell-shaped curve

Clayton is with him, takin him out of the valley.

His wife had said to him: `` Nellie is in love with Clayton Roy.

`` Exterminatin' cow thieves is just a business proposition with me '', he'd blandly announce.

It is also possible, but equally doubtful, that he actually shot down the hundreds of men with which his legend credits him.

Let me pass over the trip to Sante Fe with something of the same speed which made Mrs. Roebuck `` wonduh if the wahtahm speed limit '' ( 35 m.p.h. ) `` is still in ee-faket ''.

One of my virtues or vices is a sort of three-dimensional imagination complete with sound effects and glorious living color.

this is not so, for education offers all kinds of dividends, including how to pull the wool over a husband's eyes while you are having an affair with his wife.

It is nothing you can put your fingers on but the air suddenly fills with a high charge of electricity.

It is Eromonga -- look hard, you can see with your naked eye the wooden scaffolding on the cliff ''.

I clapped the big man with the bleached hair on his shoulder and said heartily, hoping it would make an impression on the women: `` This one is the maku Frayne.

That place is crawling with Bill Doolin and his gang ''.

The woman eyed the youth with the avidity a coin collector might display toward a rare doubloon which is not yet in his collection.

`` What is with this vow jazz ''??

-- liberal considers that the need for a national economy with controls that will assure his conception of social justice is so great that individual and local liberties as well as democratic processes may have to yield before it.

Why, in the first place, call himself a liberal if he is against laissez-faire and favors an authoritarian central government with womb-to-tomb controls over everybody??

In fact it has caused us to give serious thought to moving our residence south, because it is not easy for the most objective Southerner to sit calmly by when his host is telling a roomful of people that the only way to deal with Southerners who oppose integration is to send in troops and shoot the bastards down.

Reduced to its simplest terms, it is an assumption of a collective duty to compensate for the inability of individuals to cope with the rigors of the era.

( Since the time-span of the nation-state coincides roughly with the separate existence of the United States as an independent entity, it is perhaps natural for Americans to think of the nation as representative of the highest form of order, something permanent and unchanging.

Only one rule prevailed in my conversations with these men: The more highly placed they are -- that is, the more they know -- the more concerned they have become.

He was, and is, with the RAND Corporation, a nonprofit pool of thinkers financed by the U.S. Air Force.

They include the Navy's Atlantic Command at Norfolk, Virginia, which is in contact with the Polaris subs ; ;

In point of fact, this is a beige box with a bright red door, about one and a half feet square and hung from the wall about six feet from the door to Wisman's right.

It has nothing of the proud stride of the trained runner about it, it is not a lope, it is not done with style or verve.