Jim Coykendall, Clemson
Recently there has been much activity in the intersection of commutative algebra and graph theory. In particular, there has been much attention to the so called “zero-divisor graph” and a number of its natural variants. The idea behind the zero-divisor graph is to take as vertices the nonzero zero-divisors of a commutative ring with identity and declare that there is an edge between $x,y\in R$ if and only if $xy = 0$. There are a number of striking properties that this graph possesses, probably the most interesting of which is that this graph is connected and has diameter no more than 3.
In this talk we expand this idea to a larger arena. In particular, we fix a commutative ring and define a graph by letting the vertex set be (a subset of) the set of ideals. Under various defining conditions for edges, we determine properties of the graph that lend insights to the algebraic structures involved.