Category: Graph Theory & Applications.

Network flow

Exponential Generating Functions

Example .1 Find an exponential generating function for the number of permutations with repetition of length nn of the set {a,b,c}{a,b,c}, in which there…

Pólya–Redfield Counting

We have talked about the number of ways to properly color a graph with kk colors, given by the chromatic polynomial. For…

Binomial theorem

Binomial theorem, statement that for any positive integer n, the nth power of the sum of two numbers a and b may be expressed as the sum…

Pascal’s Identity

Pascal’s identity, first derived by Blaise Pascal in 17th century, states that the number of ways to choose k elements from…

Permutations

A permutation is an arrangement of some elements in which order matters. In other words a Permutation is an ordered Combination of…

Spanning Tree of a Graph

Spanning tree of a graph is a subgraph of it which forms a tree and contains (or spans) all the vertices…

The Inclusion-Exclusion Principle

From the First Principle of Counting we have arrived at the commutativity of addition, which was expressed in convenient mathematical notations as a + b = b + a. The Principle itself…

Rooted Trees

A rooted tree is a tree with a special vertex labelled as the “root” the of tree. The root serves as a…

Chromatic Number

Chromatic number of a graph is the minimum value of kk for which the graph is k-colorablek-colorable. In other words, it is the minimum…

Non-planar Graphs

Not all graphs are planar. If there are too many edges and too few vertices, then some of the edges…

Planar Graphs

When is it possible to draw a graph so that none of the edges cross? If this is possible, we say the…