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## Eulers polyederformula Definition: A graph is called planar if can be drawn in one plane without any arcs crossing each other.

Definintion: The graph G = (V,E) is called bipartite if the nodes can be divided into two disjunct parts V = V1∨ V2. where V1dosen’t have any elemenets in common with V2.

Eulers polyhedronformula: Let G = (V, E) be a planar, connected graph and let v denote the number of nodes, e the number of arcs and r be the number of surfaces. Then

v – e + r = 2.

Ex. For the dodecaedron, the number of surfaces is 12 similar pentagons. v = 20, e = 30 and r = 12.

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## More Graphtheory

More graph-terminology:
The distance between two nodes is the shortest distance between the two nodes.
A graph that starts and ends in the same node is called a cycle or a closed circuit.
A simple path trespasses every node only once.

Let n be a node in a graph or multigraph G. The degree or valence of v is the number of arcs having an endpoint in v.
This number can be written as deg(v).
The handshaking lemma: At a large party where everybody shakes hand but not with everybody the number of persons having shaken hand an odd number of times is even.

a graph where it is allowed to pass a node several times is called a
multi-graph.

a complete graph is a graph without loops and where every pair of nodes are connected with an arc.

Ex. Let G be a loop-free graph with n nodes, such that G has 175 arcs and its complement has 56 arcs. Determine n.
Solution: The totla number of arcs in G and its complement equals the number of arcs in the complete graph Kn.
Therefore 175 + 56 = n(n-2)/2 or ”n over 2” &imp; 231 = n(n-1)/2 &imp; n=22 ∧ n=-21.

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## Graph theory

Complicated relations between different objects/phenomena can be visualized with graphs.
a graph G is defined as an ordered pair of sets G = (V,E). where E is an ordered pair {a, b}, a,b ∈V. e.g. constitutes a pair of element in V.

The elements in V are called nodes or vertices.

The elements in are the arcs or edges of the graph. The arc e = {a,b} connects the nodes a and b or it is incident with a and b. a and b are then called end-points to the arc ab. The nodes a and b are adjacent if  ab is an arc in the graph.

Arcs starting and ending in the same node are called loops.

aan arc not incident with a node is called isolated.

Only one arc is allowed to run between two nodes.

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## Combinations

It is called acombination of r elements if the order of the elements in a selection of r elements out of n elements is irrelevant and redundance is not allowed. Another way of ststing this is to say that all elements are selected at once and not one-by-one.

This number is given by the ratio n!/(n-r)!. It is also necessary to divide by r!  since redundance is not allowed.

Ex. The capital of Madagascar is called ANTANANARIVE.

The number of letters here is 12 but to find all permutations we need to take into consideration that we have three N therefore we get the same word independently of their internal order and must theefore divide by 3!. The same is the case for the four A which force us to divide by 4!.

The number of combinations for this word therefore is 12!/(4!3!).

Generally the number of subsets with r components selected out of n elements is

C(n,r) = n!/((n-r)!r!)

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## Permutations

A permutation is an ordered arrangement of objects. Ex. If you have n objects to choose from you have n options for the first object, (n-1) options for teh second, (n-2) for the third and so on.

Therefore the number of different permutations for n objects is

n! = n(n-1)(n-2)(n-3)…….1

This is called the factorial of n!

If you wish to select r objects out of n objects this can be done in

n (n-1) (n-2)……(n-r+1) = n!/(n-r)! = P(n, r)  different ways.

Ex. The number of permutations of the 8 letters in the word SCARLET is:

8! =8* 7*6*5*4*3*2*1 = 40 320.

If oe is content with four of the letters the number of possibilities is

8!/(8-4)! = 8!/4!=1680 .

If redundance is forbidden the number of possibilitites decreases by one for every further step of the selection. The total number of possibilities for selction of r objects out of n then becomes (according to the multiplication principle)

n(n-1)(n-2) …. (n-r+1).

Kategorier If you are in a situation where you have two make two consecutive choices and the first one can be selected from n alternatives and the second can be selected from m alternatives the total number of possible combinations is n*m. This can be easilty understood since for each of the n choices of A there are m possibilities to select the second item.

Ex: Då man väljer en Golf cabriolet kan man välja mellan fyra lackfärger, fyra klädselfärger och fyra fyra motorer.
Hur många olika varianter av Golf cabriolet är teoretiskt möjliga?

Svar: 4 * 4 * 4 = 64 st.enligt multilikationsprincipen.

Ex. Determine the number of sub-sets to a set consisting of 10 elements.

A subset of a set M can be determined by going through the elementsof M, one at a time, and determines whether it belongs to the subset or not. For each element there are then 2 options: either it belongs to the subset or it doesn’t. Hence the totoal number of possibilities is 210.

Therefore the total number of subsets for the set M is 210.

In the general case when M has n elements the number of subsets is 2n.

On the other hand if you are are going to choose one item from either subset A or from subset the number of possibilities you can choose from is obviously m + n.

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## Support WWF

Kontakt
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170 81 Solna
Tel: 08-624 74 00
PG: 90 1974-6
BG: 90 1-9746
info@wwf.se

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## Peculiar functions with descriptions

The cycloid describes the motion of a point on the perimeter of a wheel.

The tractoria  (or tractrix) is a curve where the length of the tangent from the tangentialpoint to the x-axis has a constant length a. Its evolute is the chainline

y = a cosh(x/a).

a particle P attached to a string PQ pulled through Q along a given curve at right angle to thee initialposition of PQ traces out a curve called dog-curve.

The equation for this curve is x= a ln{[a+{[a+&sqrt;(a22)]/y }- &sqr;(a2-y2)

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## The Amur leopard – an ecotype Most leopards live on savannahs in tropical climate.
However at the north-eastern limit of its range it exists in a cold-temperate climate with lots of snow and down to minus 25 degrees centigrade in the winter.
Only about 40 of them remain in the wild.
Help WWF preserving this beautiful creature!