Cohen-Macaulay Circulant Graphs

Let G be the circulant graph C_n(S) with S a subset of {1,2,...,\lfloor n/2 \rfloor}, and let I(G) denote its the edge ideal in the ring R = k[x_1,...,x_n]. We consider the problem of determining when G is Cohen-Macaulay, i.e, R/I(G) is a Cohen-Macaulay ring. Because a Cohen-Macaulay graph G must be well-covered, we focus on known families of well-covered circulant graphs of the form C_n(1,2,...,d). We also characterize which cubic circulant graphs are Cohen-Macaulay. We end with the observation that even though the well-covered property is preserved under lexicographical products of graphs, this is not true of the Cohen-Macaulay property.Comment: 14 page

Shellability, vertex decomposability, and lexicographical products of graphs

In this note we describe when the independence complex of G[H], the lexicographical product of two graphs G and H, is either vertex decomposable or shellable.  As an application, we show that there exists an infinite family ofgraphs whose independence complexes are shellable but not vertexdecomposable

On Determining Minimal Spectrally Arbitrary Patterns

In this paper we present a new family of minimal spectrally arbitrary patterns which allow for arbitrary spectrum by using the Nilpotent-Jacobian method. The novel approach here is that we use the Intermediate Value Theorem to avoid finding an explicit nilpotent realization of the new minimal spectrally arbitrary patterns.Comment: 8 page