Complete characterization of graphs with exactly two positive eigenvalues.

For a graph $G$, let $\eta(G)$ denote the nullity of $G$, i.e., the multiplicity of eigenvalue zero. Denote by $\mathcal{T}_n$ the set of graphs of order $n$ such that each of them has exactly two positive eigenvalues, and $\mathcal{T}^i_n=\{G\in \mathcal{T}_n\mid \eta(G)=i\}$ for $i=0,1,2,...$. Then $\mathcal{T}_n$ can be partitioned into $\mathcal{T}_n=\mathcal{T}^0_n\cup \mathcal{T}^1_n\cup \mathcal{T}^2_n\cup\cdots$. In [Characterization of graphs with exactly two non-negative eigenvalues, Ars Math. Contemp. 12 (2017) 271--286], Oboudi characterized the graphs in $\mathcal{T}^0_n=\{G$ $\in\mathcal{T}_n\mid \eta(G)=0\}$. In [On graphs with exactly two positive eigenvalues, Ars Math. Contemp. 17 (2019) 319--347], the authors characterized the graphs in $\mathcal{T}^1_n$ $=\{G\in \mathcal{T}_n\mid \eta(G)=1\}$. To construct the graphs of $\mathcal{T}_n$ we introduce four graph transformations in this paper. Also we find 175 specific graphs selected in $\mathcal{T}_n^2$, which can not be constructed by the above graph transformations and called nonconstructible graphs. Together with 802 nonconstructible graphs selected in $\mathcal{T}_n^1$ which are listed in Table 2, we collect these 977 nonconstructible graphs in $\mathcal{U}$. Finally we show that all the other graphs of $\mathcal{T}_n$ can be constructed from the graphs in $\mathcal{T}^0_n$ and $\mathcal{U}$ by the four graph transformations. Therefore we finish the characterization of all the graphs of $\mathcal{T}_n$.