Cohen–Macaulay graphs arising from a finite commutative ring

Tóm tắt báo cáo

Let $R$ be a finite commutative ring with nonzero identity. The unitary Cayley graph (resp. unit graph) of $R$ is the graph obtained by letting all the elements of $R$ to be the vertices and defining distinct vertices $x$ and $y$ to be adjacent if and only if $x- y$ (resp. $x+y$) is a unit element of $R$. This talk is aimed at examining Cohen--Macaulayness of these graphs. Moreover, in the case $R=\mathbb Z_n$, we completely characterize Cohen--Macaulayness of the surveyed graphs. Such characterizations provide large classes of Cohen--Macaulay and non Cohen--Macaulay graphs. In addition, some issues pertaining to these graphs, such as regularity and Betti numbers, will also be presented. These stem from a joint work with T. Ashitha, T. Asir, and M. R. Pournaki.