So now let's start building
on this intuition and
let's start thinking about what is the
right graph matrix in a presentation that
will allow, allow us to use this
intuition from the previous slide, right?
So we have already defined the graph
adjacency matrix which is an n by n
matrix where n is the number of nodes.
And we simply said that a ij equals
1 if node i links to node j.
And otherwise, it has value 0.
So for a given graph G here,
this is the adjacency matrix A which
is a binary adjacency matrix where
value of 1 means that a pair of nodes
is connected and 0 means it doesn't.
What are some properties of this of
the spectrum or the eigenvalues and
eigenvectors of this adjacency matrix?
First this is a symmetric matrix,
right, because our graph is undirected.
And because of that, eigenvectors
of our adjacency matrix are real.
So are real-valued and
they are orthogonal by definition.
And these are the two important
properties that we will exploit later.
So this is now the definition
of the adjacency matrix.
Let's also define what we'll
call a degree matrix, D.
Right, so for a given graph,
given graph G,
on degree matrix D is simply
a diagonal matrix where, for
a given diagonal entry, the value there
is simply the degree of a given node.
Right, so for example, in our case.
Node number one has three edges
adjacent to it which basically means
that here in the entry 1,
1 of our degree matrix we have an n33.
This is now the definition
of the degree matrix so
now what we are ready to define is to
define the graph Laplacian matrix.
And this is simply an n by n matrix
where what we basically do is we take,
we label this matrix as L and
L is simply the degree min,
the degree matrix minus
the adjacency matrix, right?
So, now I have a matrix where on the
diagonal I have the degrees of the nodes.
And of the, of the diagonal I have
binary entries, either 0 or minus 1.
0 means that a given pair of entry,
of nodes is not connected, and minus 1
means that a pair of nodes is connected,
what are some properties of this matrix?
For example, 1 property that we
notice is that the sum of entries in
every row equals to 0, right?
Because the number of minus
1's that we have in every,
every row is exactly
the degree of the node.
And on the diagonal we have
the positive degree of that node so
that two things cancel out.
So each row and
each column of this matrix sums to 0.
What this basically means is that we have
already found a trivial eigen pair, right?
So if we have a vector
x of all values of 1.
Then if you ask, okay, I, a times x is
simply the sum of the labels of my,
of my neighbors, all my neighbors have
entry, minus 1, so I get minus d.
On the diagonal I have an entry,
d, so basically it means that l
times x equals 0, which means that.
Lambda 1 so the smallest eigenvalue
of this adjacency matrix equals 0.
And then, what are other important
properties of this Laplacian matrix?
First is that eigenvalues
are non-negative real value numbers and
the second one is, again,
that eigenvectors are real and orthogonal.
Orthogonal means that when I
do a dot product of two vec,
eigenvectors, their dot product is 0.
