Graph Laplacian Rank-One update

Question

Can anyone help me prove/disprove this conjecture?

Let $G$ be an undirected nonnegative weighted connected graph with $n$ nodes and Laplacian matrix $L$.

Also, let $0=\lambda_1<\lambda_2\leq \cdots\leq \lambda_n$ be eigenvalues of the Laplacian matrix $L$ and assume that $\max(d_i)<1$, where $d_i$ is the degree of $i^\text{th}$ node.

If $\lambda_i>1$ for some $2\leq i \leq n$, there exists an edge $e$ such that increasing weight of $e$ while having maximum degree less than one, does not increase number of eigenvalues larger than one.

edit: To leave ambiguity, $d_i = \sum_i a_{ij}$ where $a_{ij}$ is the $ij^\text{th}$ component of the Adjaceny matrix of the graph.

Answer 1

This is not a complete answer, but it can be shown that number of eigenvalues larger than $1$ can only be increase by at-most one. If you decrease the weight of an edge then number of eigenvalues larger than $1$ can only be decreased by at-most one.

This relies on the theorem that eigenvalues have interlacing property for rank one modification of symmetric matrix (see, Rank-One Modification of the Symmetric Eigenproblem by James R. Bunch 1, Christopher P. Nielsen and Danny C. Sorensen). Since, edge perturbation is a rank one-modification, the theorem follows.