I am looking for a graph for which $2 d_{i} < \mu_{i}$, for some index $i$, where $\mu_{1} \leq \mu_{2} \leq \dots\leq \mu_{n}$ are the eigenvalues of the Laplacian matrix $L(G)$ and $d_{1} \leq d_{2} \leq \dots \leq d_{n}$ are the node degrees.

According to the literature and existing upper/lower bounds on the eigenvalues of the Laplacian as a function of node degrees, it seems there is a graph with this property. However, I was not able to find such a graph by generating all graph with $4, 5, \dots, 10$ vertices. Any help or advice for a possible hypothesis confirmation or rejection would be appreciated.

Related references:

Miriam Farber, Ido Kaminer, Upper bound for the Laplacian eigenvalues of a graph, June 2011.

A. E. Brouwer, W. H. Haemers, A lower bound for the Laplacian eigenvalues of a graph—proof of a conjecture by Guo, February 2008.