Triangle-free Graphs with Three Positive Eigenvalues

A graph G is called triangle-free if G does not contain any triangle as its induced subgraph. Let G(n) be the set of triangle-free graphs of order n each of which has three positive eigenvalues. In this paper, we find 20 specific graphs in G(n), each of which has nullity no more than 2, and we show that in terms of three graph transformations all the other graphs of G(n) can be constructed from these 20 specific graphs. Hence, we completely characterize the triangle-free graphs with exactly three positive eigenvalues.