Triangle-free graphs with six non-zero eigenvalues

A graph G is called triangle-free if G does not contain a triangle as an induced subgraph. Let H-n be the set of triangle-free graphs of order n with six non-zero eigenvalues. In this paper, we find 19 graphs of H-n, and we show that the other graphs of H-n can be constructed from these 19 graphs by adding some congruent vertices. Hence we completely characterize the triangle-free graphs with six non-zero eigenvalues.