let $r$ be a commutative ring. in this paper, by using algebraic properties of $r$, we study the hase digraph of prime ideals of $r$.

Let $p$ be prime and $alpha:x mapsto xg^x$, the Discrete Lambert Map. For $kgeq 1,$ let $ V = {0, 1, 2, . . . , p^k-1}$. The iteration digraph is a directed graph with $V$ as the vertex set and there is a unique directed edge from $u$ to $alpha(u)$ for each $uin V.$ We denote this digraph by $G(g, p^{k}),$ where $gin (mathbb{Z}/p^kmathbb{Z})^*.$  In this piece of work, we investigate the struct...

let $alpha$ be an endomorphism and $delta$ an $alpha$-derivationof a ring $r$.  in this paper we  study the relationship between an$r$-module $m_r$ and the  general polynomial module  $m[x]$ over theskew polynomial ring $r[x;alpha,delta]$. we introduce the notionsof skew-armendariz modules and skew quasi-armendariz modules whichare generalizations of $alpha$-armendariz modules and extend thecla...

The energy of a digraph D with eigenvalues z1, z2, . . . , zn is defined as E(D) = n ∑ j=1 |Rzj |, where Rzj is the real part of the complex number zj . In this paper, we characterize some positive reals that cannot be the energy of a digraph. We also obtain a sharp lower bound for the energy of strongly connected digraphs.

Abstract Let D D be a digraph with vertex set V V and arc E E . For u u , the out-degree in-degree of are denoted by d + {d}_{u}^{+} − {d}_{u}^{-} respectively. A vertex-degree-based (VDB) topologic...

let $d$ be a digraph with vertex set $v(d)$. for vertex $vin v(d)$, the degree of $v$,denoted by $d(v)$, is defined as the minimum value if its out-degree and its in-degree.now let $d$ be a digraph with minimum degree $deltage 1$ and edge-connectivity$lambda$. if $alpha$ is real number, then the zeroth-order general randic index is definedby $sum_{xin v(d)}(d(x))^{alpha}$. a digraph is maximall...

For a ring endomorphism $alpha$ and an $alpha$-derivation $delta$, we introduce a concept, so called skew $pi$-Armendariz ring, that is a generalization of both $pi$-Armendariz rings, and $(alpha,delta)$-compatible skew Armendariz rings. We first observe the basic properties of skew $pi$-Armendariz rings, and extend the class of skew $pi$-Armendariz rings through various ring extensions. We nex...

 In this paper, we consider a flexible skew-generalized normal distribution. This distribution is denoted by $FSGN(/lambda _1, /lambda _2 /theta)$. It contains the normal, skew-normal (Azzalini, 1985), skew generalized normal (Arellano-Valle et al., 2004) and skew flexible-normal (Gomez et al., 2011) distributions as special cases. Some important properties of this distribution are establi...

The eigenvalues of a digraph are the eigenvalues of its adjacency matrix. The sum of the absolute values of the real part of the eigenvalues is called the energy of the digraph. The extremal energy of bicyclic digraphs with vertex-disjoint directed cycles is known. In this paper, we consider a class of bicyclic digraphs with exactly two linear subdigraphs of equal length. We find the minimal an...