Hello!

There are two definitions of graph spectrum:

1) Eigenvalues of adjacency matrix $A$.

2) Eigenvalues of Laplacian of adjacency matrix ($L$).

Different sources offer different properties based on this two definitions.
Of course it's painful to compute two different spectrums if adjacency matrix is big.
So, the question is:

Is there a method to connect one vector of eigenvalues ($\Lambda(A)$) with another ($\Lambda(L)$)?

It is obvious that

$L = T^{-1/2}(T-A)T^{-1/2} = E - T^{-1/2}AT^{-1}T^{+1/2}$, where

$T$ is the diagonal matrix with $t_{v,v}=d_v$, and $t_{u,v}=0$, if $u\ne v$,

and $t^{-1}_{v,v}=0$, if $d_v=0$,

$d_v$ - degree of $v$.

Also, when $G$ is $k$-regular, $L=I-\frac{1}{k}A$, so $\Lambda(L)=1-\frac{1}{k}\Lambda(A)$.

But in general case it's like I need to compute eigenvalues of $AT$, if I know eigenvalues of $A$. ($T$ is diagonal).

Thanks for any help.