On the nonorientable genus of the generalized unit and unitary Cayley graphs of a commutative ring

Let $R$ be a commutative ring and let $U(R)$ be multiplicative group of unit elements of $R$. In 2012, Khashyarmanesh et al. defined generalized unit and unitary Cayley graph, $\Gamma(R, G, S)$, corresponding to a multiplicative subgroup $G$ of $U(R)$ and a non-empty subset $S$ of $G$ with $S^{-1}=\{s^{-1} \mid s\in S\}\subseteq S$, as the graph with vertex set $R$ and two distinct vertices $x$ and $y$ are adjacent if and only if there exists $s\in S$ such that $x+sy \in G$. In this paper, we characterize all Artinian rings $R$ whose $\Gamma(R,U(R), S)$ is projective. This leads to determine all Artinian rings whose unit graphs, unitary Cayley garphs and co-maximal graphs are projective. Also, we prove that for an Artinian ring $R$ whose $\Gamma(R, U(R), S)$ has finite nonorientable genus, $R$ must be a finite ring. Finally, it is proved that for a given positive integer $k$, the number of finite rings $R$ whose $\Gamma(R, U(R), S)$ has nonorientable genus $k$ is finite.Comment: To appear in Algebra Colloquiu

Diameter of General Kn\"odel Graphs

The Kn\"odel graph $W_{\Delta,n}$ is a $\Delta$-regular bipartition graph on $n\ge 2^{\Delta}$ vertices and $n$ is an even integer. The vertices of $W_{\Delta,n}$ are the pairs $(i,j)$ with $i=1,2$ and $0\le j\le n/2-1$. For every $j$, $0\le j\le n/2-1$, there is an edge between vertex $(1, j)$ and every vertex $(2,(j+2^k-1) \mod (n/2))$, for $k=0,1,\cdots,\Delta-1$. In this paper we obtain some formulas for evaluating the distance of vertices of the Kn\"odel graph and by them, we provide the formula $diam(W_{\Delta,n})=1+\lceil\frac{n-2}{2^{\Delta}-2}\rceil$ for the diameter of $W_{\Delta,n}$, where $n\ge (2\Delta-5)(2^{\Delta}-2)+4$.Comment: 8 pages, 1 tabl

On the distance domination number of bipartite graphs

‎A subset D ⊆ V(G) is called a k-distance dominating set of G if every vertex in V(G)-D is within distance k from some vertex of D‎. ‎The minimum cardinality among all k-distance dominating sets of G is called the k-distance domination number of G. ‎In this note we give upper bounds on the k-distance domination number of a connected bipartite graph‎, ‎and improve some results have been given like Theorems 2.1 and 2.7 in [Tian and Xu‎, ‎A note on distance domination of graphs‎, ‎Australasian Journal of Combinatorics‎, ‎43 (2009)‎, ‎181-190]‎. </p