Let α ∈ (0,1) and let $G=(V_G,E_G$) be a graph. According to Dunbar, Hoffman, Laskar and Markus [3] a set $D\subseteq V_G$ is called an α-dominating set of G, if $|N_G\left(u\right)\cap D|\ge \alpha d_G\left(u\right)$ for all $u\in V_G\setminus D$. We prove a series of upper bounds on the α-domination number of a graph G defined as the minimum cardinality of an α-dominating set of G.

Let S be a cut of a simple connected graph G. If S has no proper subset that is a cut, we say S is a minimal cut of G. To a minimal cut S, a connected component of G-S is called a fragment. And a fragment with no proper subset that is a fragment is called an end. In the paper ends are characterized and it is proved that to a connected graph G = (V,E), the number of its ends Σ ≤ |V(G)|.

Let D be a digraph with set of vertices V and set of arcs A. We say that D is k-transitive if for every pair of vertices u, v ∈ V, the existence of a uv-path of length k in D implies that (u, v) ∈ A. A 2-transitive digraph is a transitive digraph in the usual sense. A subset N of V is k-independent if for every pair of vertices u, v ∈ N, we have d(u, v), d(v, u) ≥ k; it is l-absorbent if for every u ∈ V N there exists v ∈ N such that d(u, v) ≤ l. A k-kernel of D is a k-independent and (k − 1)-absorbent...

We study moments of the difference $D_n\left(x\right)-x^{n}n!{\mathrm{e}}^{-1/x}$ concerning derangement polynomials $D_n\left(x\right)$. For the first moment, we obtain an explicit formula in terms of the exponential integral function and we show that it is always negative for $x>0$. For the higher moments, we obtain a multiple integral representation of the order of the moment under computation.

Using isometric embedding of metric trees into Banach spaces, this paper will investigate barycenters, type and cotype, and various measures of compactness of metric trees. A metric tree (T, d) is a metric space such that between any two of its points there is a unique arc that is isometric to an interval in ℝ. We begin our investigation by examining isometric embeddings of metric trees into Banach spaces. We then investigate the possible images x₀ = π((x₁ + ... + xₙ)/n), where π is a contractive...

A signed graph (or sigraph in short) is an ordered pair $S=(S^u,\sigma )$, where $S^u$ is a graph G = (V,E), called the underlying graph of S and σ:E → +, - is a function from the edge set E of $S^u$ into the set +,-, called the signature of S. The ×-line sigraph of S denoted by $L_{\times }\left(S\right)$ is a sigraph defined on the line graph $L\left(S^u\right)$ of the graph $S^u$ by assigning to each edge ef of $L\left(S^u\right)$, the product of signs of the adjacent edges e and f in S. In this paper, first we define semi-total line sigraph and semi-total point sigraph of a given...

Let $R$ be a commutative ring. The annihilator graph of $R$, denoted by $\mathrm{AG}\left(R\right)$, is the undirected graph with all nonzero zero-divisors of $R$ as vertex set, and two distinct vertices $x$ and $y$ are adjacent if and only if ${\mathrm{ann}}_R\left(xy\right)\ne {\mathrm{ann}}_R\left(x\right)\cup {\mathrm{ann}}_R\left(y\right)$, where for $z\in R$, ${\mathrm{ann}}_R\left(z\right)=\{r\in R:rz=0\}$. In this paper, we characterize all finite commutative rings $R$ with planar or outerplanar or ring-graph annihilator graphs. We characterize all finite commutative rings $R$ whose annihilator graphs have clique number $1$, $2$ or $3$. Also, we investigate some properties of the annihilator...

Let $R$ be a ring with identity and $M$ be a unitary left $R$-module. The co-intersection graph of proper submodules of $M$, denoted by $\Omega \left(M\right)$, is an undirected simple graph whose vertex set $V\left(\Omega \right)$ is a set of all nontrivial submodules of $M$ and two distinct vertices $N$ and $K$ are adjacent if and only if $N+K\ne M$. We study the connectivity, the core and the clique number of $\Omega \left(M\right)$. Also, we provide some conditions on the module $M$, under which the clique number of $\Omega \left(M\right)$ is infinite and $\Omega \left(M\right)$ is a planar graph. Moreover, we give several...