Proving Spectrums: K6-$\lambda$I Matrix Trace

[Lauren1234]

Homework Statement

Consider the complete graph K6 (with 6 vertices).
a. Show that its spectrum is given by:
$σ(K6) = {[−1]^5, [5]}$
b. Use the spectrum given in [a.] to calculate the number of edges and triangles in K6

Relevant Equations

N/a

This is my solution so far however I’m not sure where to go from here I think it’s something to do with the trace of the matrix but. This is the full solution but I did row reduction on the matrix K6- $lambda$I

 

[StoneTemplePython]

so the adjacency matrix for $K_6 = \mathbf {11}^T - I_6 = J - I$. Can you compute the spectrum for $J$ (i.e. the all ones matrix)? Hint this is rank one, and the trace would be helpful here... after that you should know that for any matrix square matrix $B$ the new matrix $B-I$ has eigenvalues shifted down by one, why?

You can consider $\text{trace}\big(K_6^m\big)$ for well chosen $m$ for your part b.

I take it your problems are from a spectral graph theory course or perhaps general graph theory?

 

[Lauren1234]

Hi thank you I think I managed to work it out I got 15 edges and 20 triangles using a formula I found!

 

[Lauren1234]

so the adjacency matrix for $K_6 = \mathbf {11}^T - I_6 = J - I$. Can you compute the spectrum for $J$ (i.e. the all ones matrix)? Hint this is rank one, and the trace would be helpful here... after that you should know that for any matrix square matrix $B$ the new matrix $B-I$ has eigenvalues shifted down by one, why?

You can consider $\text{trace}\big(K_6^m\big)$ for well chosen $m$ for your part b.

I take it your problems are from a spectral graph theory course or perhaps general graph theory?

However the next part would you mind seeing if I’m on the correct lines it’s the same thing just for kn

 

[StoneTemplePython]

I guess I'm not really sure what you're trying to do here. I thought you were going to use spectral information to find number of edges and triangles. But you said you're using formulas you found somewhere one of which uses spectral information and one doesn't. You didn't answer what kind of class this is for...