how we can understand from spectrum of laplacian matrix of a graph that this graph is regular or not .
if we consider $0=\mu_1 \leq \mu_2 \leq ...\leq \mu_n$ as the eigenvalue of laplacian matrix ,we know that the number of spanning trees is $\kappa(G)=\frac{\mu_2\mu_3...\mu_n}{n}$.
I thought that if we have a k-regular graph ,we can obtain $\kappa(G)$ with other ways without using eigenvalues,and then we achieve a condition on eigenvalues,but I don't know if we have $\kappa(G) $ without using eigenvalues,and also I don't know this plan is right or wrong, can you give me hint and guidance about it?or any other approach to make it.thanks.
Ja, it is possible. Note that $\sum d_i$ is trace of Laplacian matrix where $d_i$ are degree of vertics. In other word, $\sum \mu_i=\sum d_i$. Now, we can compute trace $Q^2$ where $Q$ is Laplacian matrix. It is not had to see that $tr(Q^2)=\sum d_i^2+\sum d_i$.
Hence, It is enough that we check that the following equality $n\sum d_i^2=(\sum d_i)^2$.