Numpy Exercise 4

Imports



In [2]:

    
import numpy as np
%matplotlib inline
import matplotlib.pyplot as plt
import seaborn as sns

Complete graph Laplacian

In discrete mathematics a Graph is a set of vertices or nodes that are connected to each other by edges or lines. If those edges don't have directionality, the graph is said to be undirected. Graphs are used to model social and communications networks (Twitter, Facebook, Internet) as well as natural systems such as molecules.

A Complete Graph, $K_n$ on $n$ nodes has an edge that connects each node to every other node.

Here is $K_5$:



In [3]:

    
import networkx as nx
K_5=nx.complete_graph(5)
nx.draw(K_5)

The Laplacian Matrix is a matrix that is extremely important in graph theory and numerical analysis. It is defined as $L=D-A$. Where $D$ is the degree matrix and $A$ is the adjecency matrix. For the purpose of this problem you don't need to understand the details of these matrices, although their definitions are relatively simple.

The degree matrix for $K_n$ is an $n \times n$ diagonal matrix with the value $n-1$ along the diagonal and zeros everywhere else. Write a function to compute the degree matrix for $K_n$ using NumPy.



In [4]:

    
def complete_deg(n):
    """Return the integer valued degree matrix D for the complete graph K_n."""
    D = np.diag((n-1)*np.ones(n, dtype=int))
    return(D)
complete_deg(3)









    Out[4]:





array([[2, 0, 0],
       [0, 2, 0],
       [0, 0, 2]])



In [5]:

    
D = complete_deg(5)
assert D.shape==(5,5)
assert D.dtype==np.dtype(int)
assert np.all(D.diagonal()==4*np.ones(5))
assert np.all(D-np.diag(D.diagonal())==np.zeros((5,5),dtype=int))

The adjacency matrix for $K_n$ is an $n \times n$ matrix with zeros along the diagonal and ones everywhere else. Write a function to compute the adjacency matrix for $K_n$ using NumPy.



In [6]:

    
def complete_adj(n):
    """Return the integer valued adjacency matrix A for the complete graph K_n."""
    A = np.ones((n,n), dtype = int) - np.eye(n, dtype = int)
    return A



In [7]:

    
A = complete_adj(5)
assert A.shape==(5,5)
assert A.dtype==np.dtype(int)
assert np.all(A+np.eye(5,dtype=int)==np.ones((5,5),dtype=int))

Use NumPy to explore the eigenvalues or spectrum of the Laplacian L of $K_n$. What patterns do you notice as $n$ changes? Create a conjecture about the general Laplace spectrum of $K_n$.



In [28]:

    
def complete_L(n):
    K_n = nx.complete_graph(n)
    return complete_deg(n) - complete_adj(n)
for i in range(1,6):
    print np.linalg.eig(complete_L(i))[0]









    



[ 0.]
[ 2.  0.]
[  3.00000000e+00  -4.44089210e-16   3.00000000e+00]
[  4.00000000e+00  -1.11022302e-16   4.00000000e+00   4.00000000e+00]
[ 5.  0.  5.  5.  5.]



In [27]:

    
format?

n-1 eigenvalues are equal to n. The remaining eigenvalue is zero.

The eigenvector associated with the eigenvalue of zero points in the (1,1,..,1) direction (or reverse).