The Laplacian $L$ is usually defined for simple graphs, that is, graphs with no self-loops or multiple edges. Consider however the $n\times n$ matrix $M=L+P$, where $P$ has null entries except for one diagonal element $p_{jj}=1$.
Assuming that the spectrum of $L$ is known, what can we say about the spectrum of $M$? More generally, are there interesting properties that 'carry over' from one matrix to the other?
Edit 1. This is apparently called a rank-one modification, as Igor Rivin wrote below. I'm basically interested in a generalized version of this question.
You are making a rank one update of the matrix $L.$ You can read about the eigenvectors in [this Wikipedia article][1] while the eigenvalues are discussed in Gene Golub's paper:
Golub, Gene H., Some modified matrix eigenvalue problems, SIAM Rev. 15, 318-334 (1973). ZBL0254.65027.
The link to the PDF is actually in the above-mentioned Wikipedia article. [1]: https://en.wikipedia.org/wiki/Bunch%E2%80%93Nielsen%E2%80%93Sorensen_formula