Spectral characterization of mixed extensions of small graphs

A mixed extension of a graph G is a graph H obtained from G by replacing each vertex of G by a clique or a coclique, where vertices of H coming from different vertices of G are adjacent if and only if the original vertices are adjacent in G. If G has no more than three vertices, H has all but at most three adjacency eigenvalues equal to 0 or -1. In this paper we consider the converse problem, and determine the class g of all graphs with at most three eigenvalues unequal to 0 and -1. Ignoring isolated vertices, we find that g consists of all mixed extensions of graphs on at most three vertices together with some particular mixed extensions of the paths P-4 and P-5. (C) 2018 Elsevier B.V. All rights reserved.