What's pretty interesting is that the largest eigenvalue of the adjacency matrix of an undirected graph lies between the average degree and the maximum degree so the gut feeling you get when playing with the degree of a graph is legitimate.
I will jump on the opportunity do shamelessly self-plug our most recent work  on how to modify the topology of a network to have the epidemic go subcritical and quickly disappear.
The basic idea in our paper is to keep the maximum number of edges from the original graph under the constraint that the adjacency spectrum is bounded. Since that's a NP-hard problem we go for an approximation algorithm.
In any case, Melting Asphalt's essays are really an example to follow! A gold standard for expository material!
 Ganesh et al. https://ieeexplore.ieee.org/abstract/document/1498374
 Prakash et al. https://link.springer.com/article/10.1007/s10115-012-0520-y
 Bazgan et al. https://link.springer.com/chapter/10.1007/978-3-030-04651-4_...