In this note, we prove several lower bounds on the domination number of simple connected graphs. Among these are the following: the domination number is at least two-thirds of the radius of the graph, three times the domination number is at least two more than the number of cut-vertices in the graph, and the domination number of a tree is at least as large as the minimum order of a maximal matching.

The open neighborhood $N_G\left(e\right)$ of an edge $e$ in a graph $G$ is the set consisting of all edges having a common end-vertex with $e$. Let $f$ be a function on $E\left(G\right)$, the edge set of $G$, into the set $\{-1,1\}$. If ${\sum }_{x\in N_G\left(e\right)}f\left(x\right)\ge 1$ for each $e\in E\left(G\right)$, then $f$ is called a signed edge total dominating function of $G$. The minimum of the values ${\sum }_{e\in E\left(G\right)}f\left(e\right)$, taken over all signed edge total dominating function $f$ of $G$, is called the signed edge total domination number of $G$ and is denoted by ${\gamma }_{st}^{\text{'}}\left(G\right)$. Obviously, ${\gamma }_{st}^{\text{'}}\left(G\right)$ is defined only for graphs $G$ which have no connected components...

LVM and LVMB manifolds are a large family of non kähler manifolds. For instance, Hopf manifolds and Calabi-Eckmann manifolds can be seen as LVMB manifolds. The LVM manifolds have a natural action of a real torus and the quotient of this action is a polytope. This quotient allows us to relate closely LVM manifolds to the moment-angle manifolds studied by Buchstaber and Panov. Our aim is to generalize the polytope associated to a LVM manifold to the LVMB case and study the properties of this generalization....

L-zero-divisor graphs of L-commutative rings have been introduced and studied in [5]. Here we consider L-zero-divisor graphs of a finite direct product of L-commutative rings. Specifically, we look at the preservation, or lack thereof, of the diameter and girth of the L-ziro-divisor graph of a L-ring when extending to a finite direct product of L-commutative rings.