prove that we can calculate $\sum^n_{i=1} d^{2}_{i}$ from the spectrum of laplacian matrix of graph.

Question

prove that we can calculate $\sum^n_{i=1} d^{2}_{i}$ from the spectrum of laplacian matrix of graph.$d_i $are the degree of vertex $i$.

I consider bipartite graph and try to prove this result for all graphs:

consider the spectrum:$$\bigl(\begin{smallmatrix} \lambda_{1} & \lambda_{2} &... &\lambda_{n} \\ s_1& s_2 & ... & s_n \end{smallmatrix}\bigr)$$

which $s_i $s are multiplicity of $\lambda_i$s .

now I guess that the sum is $\sum _{i=1}^{n} s_{i}^2\lambda_{i}$.

now I have no Idea to how to prove this,please give me help with this ,the next I want to test more examples. thanks.

it seems that my guess is not right!

Answer 1

When you consider $Q^2$ where $Q$ is the Laplacian matrix of $G$, then you can compute trace of $Q^2$. Thus $tr(Q^2)=\sum d_i^2+\sum d_i$. In other hand, $\sum \mu_i^2-\sum d_i=\sum d_i^2$. Note that $\sum d_i=\sum \mu_i$. Hence we got $\sum d_i^2=\sum \mu_i^2-\sum \mu_i$.