$(20$ points$)$ Consider the below graph having three nodes. The weights of the respective

edges are mentioned in the graph. A function f is defined on this graph and the value

at each vertex is mentioned at the respective vertices. Here, t $=$

mod $($s$,5)+1$

$10,$ the value

of s is equal to the number of characters in your first name and mod represents the

modulus operator.

$0\mathrm{.}80\mathrm{.}1$

$0\mathrm{.}3$

t

t

$($a$)$ Find the Laplacian matrix in its eigenvalues for the given graph.

$($b$)$ Define a filter as g$(\backslash$Lambda $)=$ I$+$t$\backslash$Lambda $\backslash$Lambda

$2$ on this graph. Here, I is the identity matrix, $\backslash$Lambda

is a diagonal matrix containing the eigenvalues of the Laplacian matrix of the given

graph. Find the convolution of the signal f and filter g on the graph.