I am reading the book "Spectra of Graphs" by Andries E. Brouwer and Willem H. Haemers, in which the graphs $ \widehat{E}_n $ is referenced, but not given for every n.
When Showing some examples, for n = 6,7,8 they look like this (ignore the numbers near the vertecies):
I would love to know if there is a general description of this graph.