In an assignment, I wrote about Isomorphism. Isomorphism in linear algebra, in Adjacency Matrix, In Graph Vector.

1. Isomorphism
By Mahe Karim
ID: 162-15-7770
Sec: B
Dept. Of CSE
Daffodil International University

2. What is Isomorphism ?
 In any category C, an arrow f : A → B is called Isomorphism.
 Two graphs, G=(V,E,I) and H=(W,F,J), are isomorphic (normally written in the
form G=H, where the = should have a third wavy line above the the two
parallel lines), if there are bijections f:V->W and g:E->F such that eIv if and
only if g(e)Jf(v). Two isomorphic graphs must have exactly the same set of
parameters. For example, the cardinalities of the vertex sets must be equal,
the cardinalities of the edge sets must be equal, the (ordered) degree
sequences must be the same, any graph polynomials must agree on the two
graphs

3. Applications
 In abstract algebra, two basic isomorphisms are defined:
 Group isomorphism, an isomorphism between groups
 Ring isomorphism, an isomorphism between rings.
 Just as the automorphisms of an algebraic structure form a group, the isomorphisms
between two algebras sharing a common structure form a heap. Letting a particular
isomorphism identify the two structures turns this heap into a group.
 In mathematical analysis, the Laplace transform is an isomorphism mapping hard
differential equations into easier algebraic equations.
 In category theory, let the category C consist of two classes, one of objects and the
other of morphisms. Then a general definition of isomorphism that covers the
previous and many other cases is: an isomorphism is a morphism ƒ: a → b that has an
inverse, i.e. there exists a morphism g: b → a with ƒg = 1b and gƒ = 1a. For example,
a bijective linear map is an isomorphism between vector spaces, and a bijective
continuous function whose inverse is also continuous is an isomorphism between
topological spaces, called a homeomorphism.

4. Isomorphic Graph
 Two graphs which contain the same number of graph vertices connected in the
same way are said to be isomorphic. Formally, two graphs and with graph vertices
are said to be isomorphic if there is a permutation of such that is in the set of
graph edges if is in the set of graph edges .

5. Isomorphism - Linear Algebra
 Two vector spaces are isomorphic if there is an invertible linear
transformation between them. Any such invertible linear transformation is
an isomorphism. In particular, if a vector space has a finite basis, then it
is isomorphic to the euclidean space. ... whose matrix has the given basis
vectors as columns.

6. Examples of isomorphism

7. Real Life Applications Of Isoorphism
 Here are a couple:
 Paths through a perfectly rectangular lattice city (taxicab metric) biject to
rearrangements of two letters like AABBBABAABA. (Let A=step south and
B=step west if your destination lies to the south-west.)
 Friend graphs like in Facebook biject to adjacency matrices. The matrix
would be different if they are weighted (e.g. EdgeRank) versus unweighted,
directly vs undirected.
 The group of rotations of Erno Rubik's cube bijects to quantum
chromodynamic rules that govern quarks making atoms.

8. Reference
 *
https://en.wikibooks.org/wiki/Linear_Algebra/Definition_and_Examples_of_I
somorphisms
 * http://algebra.math.ust.hk/vector_space/08_basis/lecture3.shtml
 * http://math.fau.edu/locke/Graphmat.htm
 * http://maths.isomorphism.es