Toric Fano varieties associated to finite simple graphs (Q2418805)

For a finite simple graph \(G\), one can construct a normal fan \(\Delta(G)\) of the graph associahedron of \(G\). Then the toric variety \(X(\Delta(G))\) associated to the fan \(\Delta(G)\) is smooth and projective. This construction is reviewed in Section 2 of the paper under review. The first main result of the paper is that \(X(\Delta(G))\) is Fano if and only if each connected component of \(G\) has at most three nodes (Theorem 3.1). The second main result of the paper is a similar characterization when \(X(\Delta(G))\) is weak Fano in terms of the graph (Theorem 3.4).