Triangle-free Graphs with Three Positive Eigenvalues

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