Fun Facts about Networks

Luisa Cutillo

l.cutillo@leeds.ac.uk

Networks

Technological

Social

Biological

Real Data Application

Networks in maths

Networks are mathematical representations of interactions among the components of a system and can be modelled by

graphs

  • Components: Nodes, Vertices
  • Interactions: Links, Edges
  • Systems: Network, Graph              
  • V
  • E
  • G=(V,E)              

 A planar graph is one in which the edges have no intersection or common points except at their endpoints.

 

  • planar graph
  • non planar
  • complete
  • both connected

 Euler had discovered something peculiar about calculating together the number of vertices, edges and regions

V=number of vertices= 5

E=number of edges= 6

R=number of regions= 3 (the outside region counts!)

 

 

 

 

V-E+R=2

Let's try again

V=number of vertices= 5

E=number of edges= 7

R=number of regions= 4 (the outside region counts!)

 

 

 

 

V-E+R=2

Let's try again

V-E+R=2

 

 

 

 

 

 

?

 

 

 

 

 

Is it always true?

V-E+R\ne2

 

 

 

 

 

 

?

 

 

 

 

 

V-E+R=2

Planar!

Is it always true?

V-E+R\ne2
V-E+R=2

Connected!

The Euler characteristic:

  V−E+R

is always 2  in the case of connected planar graphs

Conclusion

A proof by induction is a mathematical proof technique:

1- base case, proves the statement for n = 0 without assuming any knowledge of other cases. 
2-induction step, proves that if the statement holds for any given case n = k, then it must also hold for the next case n = k + 1. 

These two steps establish that the statement holds for every natural number n. 
(*base case not necessarily begin with 0, but often with 1 or any fixed number N)

Proof by induction

Proof by induction

V= 1

E=  0

R=  1

 

 

V= 1+1=2

E=  0+1=1

R=  1

 

 

  • true when adding one new vertex connected by one new edge
  • true when carving out a new region from part of an old region with the addition of a single new edge
  • Since any connected planar graph can be built up in this way by gradually adding new vertices and edges, this argument shows V−E+R=2 for any connected planar graph.

Vn= 3

En=  3

R=  2

Vn+1=Vn +1

En+1=En +1 

Rn+1= Rn 

Vn+2=Vn+1

En+2=En+1 +1

Rn+2= Rn+1 +1

V-E+R=2 TRUE!

Eulerian Paths

 A route that never passes over an edge more than once, allowing for revisiting vertices, is called a Eulerian path.

An Eulerian circuit starts and ends at the same vertex, but an Eulerian path can start and end at different vertices.

Circuit

Path

Eulerian Paths and Circuits

Circuit

Path

  • even number of edges at each vertex

(even degree)

  • the starting and ending vertices have odd degree
  • the other vertices have even degree

Euler’s assertion that a graph possessing such a path has at most two vertices of odd degree was the first theorem in graph theory.

Problem: 7 bridges of konigsberg 

Could a person leave home, take a walk, and return, crossing each bridge just once?

A

B 

C

D

Problem: 

Could a person leave home, take a walk, and return, crossing each bridge just once?

This graph has more than 2 odd vertices!

A

B

 

C

D

 

A

B 

C

D

Not Eulerian! 

Impossible problem!

5 rooms problem:

can you cross each "wall" of the diagram with a continuous line only once?

Odd

Even

 

 

 

?

 

 

 

 

 

Thank You!