Maths - Graph Theory - Martin Baker

Sum, Product and Exponent of Graphs

There are many operations that we can apply to graphs including:

Sum - disjoint union
Product - at least two types of product
- Cartesian product
- tensor product
Exponent - function space.

	Nodes	Arcs
Sum disjoint union	The nodes are added in the same way as sum (disjoint union) of sets.	Arcs are just moved with the corresponding nodes.
Product Cartesian	We can take the Cartesian product of two domains A and B.	If: a₁->a₂ and b₁->b₂ then: <a₁,b₁>-><a₂,b₂>
Product tensor	We can take the Cartesian product of two domains A and B.	If: a₁->a₂ and b₁->b₂ then: <a₁,b₁>-><a₂,b₂>
Exponent (function)		For f,gHom(A,B) f->g if f(x)->g(x) for all xA

This page contains the following examples:

Simple Example
Non-simple Example
Check Product Axioms
Check Using Adjacency Matrix

Simple Example

First we will try out some sums and products of very simple graphs, just to get a feel for how things work.

Here we construct a very simple graph with two nodes "a" and "b" with an arrow between them:

Graph

Adjacency Matrix

	a	b
a	0	1
b	0	0

Now we build the other operand for our various sums and products. This will also be a simple graph with two nodes and an arrow between them, but we will draw it differently and use different names to make the resulting diagrams more useful.

Graph

Adjacency Matrix

	1	2
1	0	1
2	0	0

Now we have two graphs to combine we can start testing out the sums and products. First a sum (which uses the "+" operator).

As you can see below this produces a graph with 4 vertices and 2 arrows. The arrows do not interact.

Graph

Adjacency Matrix

	b	2
a	1	0
b	0	0
1	0	1
2	0	0

We can now go on to to look at 'merge', this is similar to '+' above, in fact if we used the same operands as above it would have produced the same result. In order to illustrate the effect of 'merge' we will change the second operand slightly, it now has vertex 'a' and vertex '2' so it shares 'a' with the first operand. As we can see below, this common vertex is used by both arrows:

Graph

Adjacency Matrix

	b	2
a	1	1
b	0	0
2	0	0

We can now go on to look at the types of product supported. There are two types of product:

Tensor Product
Cartesian Product

First the tensor product, this generates 4 vertices (2*2) and one arrow which is a sort of composition of the two arrows from the two operands. The tensor product is represented by the '*' operation:

Graph

Adjacency Matrix

	a1	b1	a2	b2
a1	0	0	0	1
b1	0	0	0	0
a2	0	0	0	0
b2	0	0	0	0

Now the Cartesian product, this always has the same vertices as the tensor product. However the arrows are different:

Graph

Adjacency Matrix

	b1	a2	b2
a1	1	1	0
b1	0	0	1
a2	0	0	1
b2	0	0	0

Non-simple Example

Now that we have seen a simple case of these operations we will look at a more complicated case for the products. So that we can check our results we will use the examples on these wiki pages:

Here we construct the first operand from the wiki pages:

Graph

Adjacency Matrix

	b	c	d	e
a	1	1	0	0
b	0	0	1	0
c	0	0	1	0
d	0	0	0	1
e	0	0	0	0

Now we build the other operand from the wiki pages.

Graph

Adjacency Matrix

	2	3
1	1	0
2	0	1
3	0	0
4	0	0

We can now go on to look at the types of product supported. There are two types of product:

Tensor Product
Cartesian Product

First the tensor product:

Graph

Adjacency Matrix

	b2	c2	d2	e2	b3	c3	d3	e3	b4	c4	d4	e4
a1	1	1	0	0	0	0	0	0	0	0	0	0
b1	0	0	1	0	0	0	0	0	0	0	0	0
c1	0	0	1	0	0	0	0	0	0	0	0	0
d1	0	0	0	1	0	0	0	0	0	0	0	0
e1	0	0	0	0	0	0	0	0	0	0	0	0
a2	0	0	0	0	1	1	0	0	1	1	0	0
b2	0	0	0	0	0	0	1	0	0	0	1	0
c2	0	0	0	0	0	0	1	0	0	0	1	0
d2	0	0	0	0	0	0	0	1	0	0	0	1
e2	0	0	0	0	0	0	0	0	0	0	0	0
a3	0	0	0	0	0	0	0	0	0	0	0	0
b3	0	0	0	0	0	0	0	0	0	0	0	0
c3	0	0	0	0	0	0	0	0	0	0	0	0
d3	0	0	0	0	0	0	0	0	0	0	0	0
e3	0	0	0	0	0	0	0	0	0	0	0	0
a4	0	0	0	0	1	1	0	0	0	0	0	0
b4	0	0	0	0	0	0	1	0	0	0	0	0
c4	0	0	0	0	0	0	1	0	0	0	0	0
d4	0	0	0	0	0	0	0	1	0	0	0	0
e4	0	0	0	0	0	0	0	0	0	0	0	0

Now the Cartesian product:

Graph

Adjacency Matrix

	b1	c1	d1	e1	a2	b2	c2	d2	e2	a3	b3	c3	d3	e3	a4	b4	c4	d4	e4
a1	1	1	0	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0
b1	0	0	1	0	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0
c1	0	0	1	0	0	0	1	0	0	0	0	0	0	0	0	0	0	0	0
d1	0	0	0	1	0	0	0	1	0	0	0	0	0	0	0	0	0	0	0
e1	0	0	0	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	0
a2	0	0	0	0	0	1	1	0	0	1	0	0	0	0	1	0	0	0	0
b2	0	0	0	0	0	0	0	1	0	0	1	0	0	0	0	1	0	0	0
c2	0	0	0	0	0	0	0	1	0	0	0	1	0	0	0	0	1	0	0
d2	0	0	0	0	0	0	0	0	1	0	0	0	1	0	0	0	0	1	0
e2	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	0	0	0	1
a3	0	0	0	0	0	0	0	0	0	0	1	1	0	0	0	0	0	0	0
b3	0	0	0	0	0	0	0	0	0	0	0	0	1	0	0	0	0	0	0
c3	0	0	0	0	0	0	0	0	0	0	0	0	1	0	0	0	0	0	0
d3	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	0	0	0	0
e3	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	1	0	0
b4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0
c4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0
d4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1
e4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0

Check Product Axioms

Here I would like to investigate the properties of the tensor and Cartesian products of graph. So is it associative? First we need to create 3 elements of graph which are not too big but just big enough to be interesting:

Here is the first graph:

(47) -> hs21 := ["a","b","c"]::GS
   (47)  "a","b","c"|"a1":"a"->"b","a2":"b"->"c"
                                            Type: DirectedGraph(String)

(48) -> diagramSvg("testGraph/testGraph4s12.svg",hs21,true)
                                                             Type: Void

Here is the second graph:

(49) -> hs22 := cycleClosed(["1","2","3"],"b")$GS
   (49)  "1","2","3"|"b1":"1"->"2","b2":"2"->"3","b3":"3"->"1"
                                            Type: DirectedGraph(String)

(50) -> diagramSvg("testGraph/testGraph4s13.svg",hs22,true)
                                                             Type: Void

Here is the third graph:

(51) -> hs23 := ["x","y","z"]::GS
   (51)  "x","y","z"|"a1":"x"->"y","a2":"y"->"z"
                                            Type: DirectedGraph(String)

(52) -> diagramSvg("testGraph/testGraph4s14.svg",hs23,true)
                                                             Type: Void

First we will try tensor product.

Here is (hs21*hs22)*hs23:

Graph

Adjacency Matrix

	b1x	c1x	b2x	c2x	b3x	c3x	b1y	c1y	b2y	c2y	b3y	c3y	b1z	c1z	b2z	c2z	b3z	c3z
a1x	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
b1x	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
c1x	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a2x	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
b2x	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
c2x	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a3x	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
b3x	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
c3x	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a1y	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
b1y	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
c1y	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a2y	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
b2y	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
c2y	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a3y	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
b3y	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
c3y	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a1z	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
b1z	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
c1z	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a2z	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
b2z	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
c2z	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a3z	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
b3z	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
c3z	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0

and here is hs21*(hs22*hs23) :

Graph

Adjacency Matrix

	b1x	c1x	b2x	c2x	b3x	c3x	b1y	c1y	b2y	c2y	b3y	c3y	b1z	c1z	b2z	c2z	b3z	c3z
a1x	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
b1x	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
c1x	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a2x	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
b2x	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
c2x	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a3x	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
b3x	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
c3x	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a1y	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
b1y	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
c1y	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a2y	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
b2y	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
c2y	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a3y	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
b3y	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
c3y	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a1z	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
b1z	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
c1z	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a2z	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
b2z	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
c2z	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a3z	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
b3z	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
c3z	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0

'looseEquals' returns true, so the two graphs are isomorphic, which means that (hs21*hs22)*hs23 is isomorphic to hs21*(hs22*hs23) where '*' is tensor product.

Now we can try Cartesian product.

Here is (hs21*hs22)*hs23:

Graph

Adjacency Matrix

	b1x	c1x	b2x	c2x	b3x	c3x	b1y	c1y	b2y	c2y	b3y	c3y	b1z	c1z	b2z	c2z	b3z	c3z
a1x	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
b1x	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
c1x	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a2x	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
b2x	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
c2x	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a3x	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
b3x	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
c3x	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a1y	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
b1y	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
c1y	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a2y	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
b2y	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
c2y	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a3y	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
b3y	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
c3y	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a1z	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
b1z	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
c1z	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a2z	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
b2z	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
c2z	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a3z	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
b3z	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
c3z	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0

and here is hs21*(hs22*hs23) :

Graph

Adjacency Matrix

	b1x	c1x	b2x	c2x	b3x	c3x	b1y	c1y	b2y	c2y	b3y	c3y	b1z	c1z	b2z	c2z	b3z	c3z
a1x	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
b1x	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
c1x	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a2x	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
b2x	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
c2x	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a3x	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
b3x	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
c3x	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a1y	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
b1y	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
c1y	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a2y	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
b2y	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
c2y	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a3y	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
b3y	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
c3y	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a1z	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
b1z	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
c1z	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a2z	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
b2z	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
c2z	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
a3z	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
b3z	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
c3z	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0

'looseEquals' returns false, however the diagrams look the same except some of the arrow names are different. So the graphs may well be isomorphic? We really need a proper test for isomorphism between graphs.

Check Using Adjacency Matrix

We can calculate the adjacency matrix of our required product by taking:

Cartesian product: by taking Kronecker sum of adjacency matrix.
Tensor product: by taking Kronecker product of adjacency matrix.

(63) -> am21 := adjacencyMatrix(hs21)
         +0  0  0+
   (63)  |1  0  0|
         +0  1  0+
                                       Type: Matrix(NonNegativeInteger)

(64) -> am22 := adjacencyMatrix(hs22)
         +0  0  1+
   (64)  |1  0  0|
         +0  1  0+
                                       Type: Matrix(NonNegativeInteger)

(65) -> hs28 := closedTensor(hs21,hs22,concat)$GS
   (65)
   "a1","a2","a3","b1","b2","b3","c1","c2","c3"|"a1*b1":"a1"->"b2",
     "a1*b2":"a2"->"b3",
     "a1*b3":"a3"->"b1",
     "a2*b1":"b1"->"c2",
     "a2*b2":"b2"->"c3",
     "a2*b3":"b3"->"c1"
                                            Type: DirectedGraph(String)

(66) -> am28 := adjacencyMatrix(hs28)
         +0  0  0  0  0  0  0  0  0+
         |0  0  0  0  0  0  0  0  0|
         |0  0  0  0  0  0  0  0  0|
         |0  0  1  0  0  0  0  0  0|
   (66)  |1  0  0  0  0  0  0  0  0|
         |0  1  0  0  0  0  0  0  0|
         |0  0  0  0  0  1  0  0  0|
         |0  0  0  1  0  0  0  0  0|
         +0  0  0  0  1  0  0  0  0+
                                       Type: Matrix(NonNegativeInteger)

(67) -> kroneckerProduct(am21,am22)
         +0  0  0  0  0  0  0  0  0+
         |0  0  0  0  0  0  0  0  0|
         |0  0  0  0  0  0  0  0  0|
         |0  0  1  0  0  0  0  0  0|
   (67)  |1  0  0  0  0  0  0  0  0|
         |0  1  0  0  0  0  0  0  0|
         |0  0  0  0  0  1  0  0  0|
         |0  0  0  1  0  0  0  0  0|
         +0  0  0  0  1  0  0  0  0+
                                       Type: Matrix(NonNegativeInteger)

(68) -> hs29 := closedCartesian(hs21,hs22,concat)$GS
   (68)
   "a1","a2","a3","b1","b2","b3","c1","c2","c3"|"b1#1":"a1"->"a2",
     "b2#1":"a2"->"a3","b3#1":"a3"->"a1","a1#1":"a1"->"b1",
     "a1#2":"a2"->"b2","a1#3":"a3"->"b3","b1#2":"b1"->"b2",
     "b2#2":"b2"->"b3","b3#2":"b3"->"b1","a2#1":"b1"->"c1",
     "a2#2":"b2"->"c2","a2#3":"b3"->"c3","b1#3":"c1"->"c2",
     "b2#3":"c2"->"c3","b3#3":"c3"->"c1"
                                            Type: DirectedGraph(String)

(69) -> am29 := adjacencyMatrix(hs29)
         +0  0  1  0  0  0  0  0  0+
         |1  0  0  0  0  0  0  0  0|
         |0  1  0  0  0  0  0  0  0|
         |1  0  0  0  0  1  0  0  0|
   (69)  |0  1  0  1  0  0  0  0  0|
         |0  0  1  0  1  0  0  0  0|
         |0  0  0  1  0  0  0  0  1|
         |0  0  0  0  1  0  1  0  0|
         +0  0  0  0  0  1  0  1  0+
                                       Type: Matrix(NonNegativeInteger)

(70) -> kroneckerSum(am21,am22)
         +0  0  1  0  0  0  0  0  0+
         |1  0  0  0  0  0  0  0  0|
         |0  1  0  0  0  0  0  0  0|
         |1  0  0  0  0  1  0  0  0|
   (70)  |0  1  0  1  0  0  0  0  0|
         |0  0  1  0  1  0  0  0  0|
         |0  0  0  1  0  0  0  0  1|
         |0  0  0  0  1  0  1  0  0|
         +0  0  0  0  0  1  0  1  0+
                                       Type: Matrix(NonNegativeInteger)

That concludes our overview of ways to combine graphs such as sums and products graphs, click here to go to the next stage of the tutorial which is about ways to create new graphs by mapping from existing graphs.

metadata block

see also:	For a discussion about using and representing graphs in category theory see this page. There is a program which can represent and do calculations on graph theory on this page.
Correspondence about this page
Book Shop - Further reading. Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.	Modern Graph Theory (Graduate Texts in Mathematics, 184)
Terminology and Notation Specific to this page here:

Maths - Graph Theory - Product

Sum, Product and Exponent of Graphs

Simple Example

Non-simple Example

Check Product Axioms

Check Using Adjacency Matrix

EuclideanSpace

Prerequisites

Computing Graphs

Problem Solving