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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.
Fang Duan (Thu,) studied this question.