The parameter q(G) of a graph G is the minimum number of distinct eigenvalues over the family of symmetric matrices described by G. It is shown that the minimum number of edges necessary for a connected graph G to have q(G)=2 is 2n−4 if n is even, and 2n−3 if n is odd. In addition, a characterization of graphs for which equality is achieved in either case is given.
@article{arxiv.2206.08860,
title = {Sparsity of Graphs that Allow Two Distinct Eigenvalues},
author = {Wayne Barrett and Shaun Fallat and Veronika Furst and Franklin Kenter and Shahla Nasserasr and Brendan Rooney and Michael Tait and Hein van der Holst},
journal= {arXiv preprint arXiv:2206.08860},
year = {2022}
}