upsc mathematics optional paper

Definition :  Two graphs G and H are isomorphic, denoted by G ≅ H, if there exists a bijection α:Vg➝Vh such that :                                   uv ∈ Eg  ⇔ α(u)α(v) ∈ Eh ; for all u,v ∈ G Hence G and H are isomorphic if the vertices of H are remaining of those of G.Two isomorphic graphs enjoy the same graph-theoretical properties and they are often identified.In particular,all isomorphic graphs have the same plane figures (excepting the identities of the vertices). This shows in the figures, where we tend to replace the vertices by small circles and talk of 'the graph' although there are, in fact, infinitely many such graphs. Example 1 : The following graphs are isomorphic . Indeed, the required isomorphism is given by                                                     v1 ↦1, v2 ↦ 3,v3 ↦ 4,v4 ↦2, v5 ↦5. Isomorphism Problem : Does there exist an efficient algorithm to check whether any two given graphs are isomorphic or not? The following table lis

Introduction : Graph theory may be said to have its beginning in 1736 when Euler considered the (general case of the) Konigsberg bridge problem. Does there exist a walk crossing each of the seven bridges of Konigsberg exactly once? It took 200 years before the first book on graph theory was written. This was " Theorie der endlichen und unendlichen Graphen " by koing in 1936 .Since,then graph theory has developed into an extensive and popular branch of mathematics,which has been applied to many problems in mathematics, computer science and other scientific and not-s-scientific areas. There are no standard notations for graph theoretical objects. This is natural because the names one uses for the objects reflect the applications. Thus, for instance, if we consider a communication network (say, for email) as a graph, the computers taking part in this network, are called nodes rather than vertices or points.On the other hand, other names are used for molecular structures

T uple  =  A tuple is a finite ordered list of elements sequence of n-elements where n is a non-negative integer. Set of n-tuples of Real Numbers =>   R n Note:  O-tuple = Empty space  Eg. (0,0,0,0...........) A   particular n tuple in  R n  U =  (  a1,a2,a3,..................an  ) is called a point or vector. [   where, (a1,a2,a3,.............an) is called  Scalar  & U is called  Vector. ]                                            or, U = [a¡] ,where i is called as components/entries/co-ordinates/elements. Note:  Elements of  R =  S calar . 1.) Two vectors U & V are equal i.e U=V if they have same number of components and the corresponding components are equal .  e.g.  U =[a1,a2,a3]  V =[b1,b2,b3]  If U =V then    a1=b1 , a2=b2 , a3=b3 2.) U=[0,0,0..........0] This is Zero vector and is denoted as Bold 0 (𝐨). Example : Q ) Identify the space :  a) (2,-5) ->  R 2  b) (3,4,5) -> R^3 Q ) Find x,y,z such that (x-y , x+y , Z-

When we have to calculate the factorial of a non-negative integer n, then it is denoted by n!, is the product of all positive integer less than or equal to n . so, 𝑛 ! = 𝑛 ∗ ( 𝑛 − 1 ) ∗ ( 𝑛 − 2 ) ∗ ( 𝑛 − 3 ) ∗ . . .3 ∗ 2 ∗ 1 n ! = n ∗ ( n − 1 ) ∗ ( n − 2 ) ∗ ( n − 3 ) ∗ . . .3 ∗ 2 ∗ 1 However, the recursive definition of factorial is of more use in this proof. 𝑛 ! = { 1 𝑛 ∗ ( 𝑛 − 1 ) ! 𝑛 = 0 𝑛 > 0 n ! = { 1 n = 0 n ∗ ( n − 1 ) ! n > 0 The recursive definition of  Factorial  leads to one interesting way of expressing factorial numbers . 𝑛 ! = ( 𝑛 + 1 ) ! ( 𝑛 + 1 ) n ! = ( n + 1 ) ! ( n + 1 ) This is valid since, as we expand  ( 𝑛 + 1 ) ! ( n + 1 ) !  from the recursive definition, we can cancel  ( 𝑛 + 1 ) ( n + 1 )  term from both numerator and denominator to get  𝑛 ! n ! . Or we can even calculate factorial in the numerator and then evaluate the division For example, 5 ! = 6 ! 6 = 720 6 = 4 ! = 5 ! 5 = 120 5 3 ! = 4 ! 4 = 24 4 2 ! = 3 ! 3