Normal Polytopes Arising from Finite Graphs

LetGbe a finite connected graph on the vertex set {1,…,d} allowing loops and having no multiple edge. LetK[t1,…,td] denote the polynomial ring indindeterminates over a fieldKand letK[G] be the subalgebra ofK[t1,…,td] generated by all quadratic monomialstitjsuch that {i, j} is an edge ofGand by all quadratic monomialst2isuch thatGhas a loop ati. We describe the normalization ofK[G] explicitly and we give a combinatorial criterion forK[G] to be normal.