# Connection between eigenvalues of matrix and its Laplacian.

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.

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Essentially your question is equivalent to asking for the relation between the spectrum of $A+D$ and $A$, where are $A$ is symmetric, $D$ is diagonal and both matrices are real. And, by change of basis, this is equivalent to asking for between the spectrum of $A+B$ and $A$ when $A$ and $B$ are real symmetric. A lot of thought has been given to this question. The short summary is that eigenvalues of $A$ provide no information useful towards computing the eigenvalues of $A+D$.