Serre's Properties for Quadratic Generated Domains from Graphs

For any graph, one can construct a ring, called the edge ring, which is a quadratic-monomial generated subring of the Laurent polynomial ring $k[x_1^{\pm 1},\dots,x_n^{\pm 1}]$. In fact, every quadratic-monomial generated subring of this Laurent polynomial ring can be generated as an edge ring for some graph. The combinatorial structure of the graph has been successfully applied to identify and classify many important commutative algebraic properties of the corresponding edge ring. In this paper, we classify Serre's $R_1$ condition for all quadratic-monomial generated subrings of $k[x_1^{\pm 1},\dots,x_n^{\pm 1}]$. Moreover, we provide a minimal example of a graph whose corresponding edge ring is not Cohen-Macaulay. This paper extends the work of Hibi and Ohsugi from the setting subrings of polynomial rings to subrings of Laurent polynomial rings.