A Laplacian matrix $L\in\mathbb{R}^{n\times n}$, is a symmetric matrix with entries, \begin{equation} l_{ij}=\begin{cases} 1=\sum_{i,~ i\neq j} w_{ij} &\mbox{if } i=j \\ -w_{ij} & \mbox{otherwise},\end{cases} \end{equation} ie, off-diagonal entries, which are negative, are summing, in absolute value, to $1$ which resides on the diagonal. Note that the diagonal entries are the only positive entries. From the Gershgorin theorem it follows that the spectra of $L$ is within the range $[0, 2]$, but it is possible to obtain a tighter bound, ie, $[0, 2)$.
Now, suppose a pseudo-Laplacian $K$ is given, defined similarly as $L$, but with $w'_{ij}$ corresponding to arbitrary numbers summing, in absolute value, to diagonal entries $k_{ii}$, $\forall i$, \begin{equation} k_{ij}=\begin{cases} \sum_{i,~ i\neq j} w'_{ij} &\mbox{if } i=j \\ -w'_{ij} & \mbox{otherwise}.\end{cases} \end{equation} So, for both $L$ and $K$, the off-diagonal entries are negative ($w_{ij}$ and $w'_{ij}$ are positive) Let $D=diag(K)$. Can it be shown that the spectrum of $D^{-1}K$ is within the range $[0, 2)$?
My primary intention was to derive a connection between $K$ and $L$, and to then establish spectrum equivalence. Note that for certain $L'\subset L$, $n>2$, having zero off-diagonal entries, the spectrum range includes $2$, ie, $[0, 2]$. A connection between $K$ and $L$ would therefore help establishing a bound on $D^{-1}K$, with $K$ possibly having some zero off-diagonal entries.