A note on acyclic domination number in graphs of diameter two

Author name used in this publication: C. T. Ng2005-2006 > Academic research: refereed > Publication in refereed journalAccepted ManuscriptPublishe

The total zero-divisor graph of commutative rings

In this paper we initiate the study of the total zero-divisor graphs over commutative rings with unity. These graphs are constructed by both relations that arise from the zero-divisor graph and from the total graph of a ring. We characterize Artinian rings with the connected total zero-divisor graphs and give their diameters. Moreover, we compute major characteristics of the total zero-divisor graphs of the ring ${\mathbb Z}_m$ of integers modulo $m$ and prove that the total zero-divisor graphs of ${\mathbb Z}_m$ and ${\mathbb Z}_n$ are isomorphic if and only if $m=n$

On equality in an upper bound for the acyclic domination number

A subset $A$ of vertices in a graph $G$ is acyclic if the subgraph it induces contains no cycles. The acyclic domination number $\gamma_a(G)$ of a graph $G$ is the minimum cardinality of an acyclic dominating set of $G$. For any graph $G$ with $n$ vertices and maximum degree $\Delta(G)$, $\gamma_a(G) \leq n - \Delta(G)$. In this paper we characterize the connected graphs and the connected triangle-free graphs which achieve this upper bound

Vertex decomposable graphs, codismantlability, Cohen-Macaulayness and Castelnuovo-Mumford regularity

We call a (simple) graph G codismantlable if either it has no edges or else it has a codominated vertex x, meaning that the closed neighborhood of x contains that of one of its neighbor, such that G-x codismantlable. We prove that if G is well-covered and it lacks induced cycles of length four, five and seven, than the vertex decomposability, codismantlability and Cohen-Macaulayness for G are all equivalent. The rest deals with the computation of Castelnuovo-Mumford regularity of codismantlable graphs. Note that our approach complements and unifies many of the earlier results on bipartite, chordal and very well-covered graphs