let $g$ be a simple graph‎, ‎and $g^{sigma}$‎ ‎be an oriented graph of $g$ with the orientation ‎$sigma$ and skew-adjacency matrix $s(g^{sigma})$‎. ‎the $k-$th skew spectral‎ ‎moment of $g^{sigma}$‎, ‎denoted by‎ ‎$t_k(g^{sigma})$‎, ‎is defined as $sum_{i=1}^{n}( ‎‎‎lambda_{i})^{k}$‎, ‎where $lambda_{1}‎, ‎lambda_{2},cdots‎, ‎lambda_{n}$ are the eigenvalues of $g^{sigma}$‎. ‎suppose‎ ‎$g^{sigma...

We consider the system of equations $A_k(x)=p(x)A_{k-1}(x)(q(x)+\sum_{i=0}^k A_i(x))$ for $k\geq r+1$ where $A_i(x)$, $0\leq i \leq r$, are some given functions and show how to obtain a close form $A(x)=\sum_{k\geq 0}A_k(x)$. apply this general result enumeration certain subsets Dyck, Motzkin, skew Motzkin paths, defined recursively according first return decomposition with monotonically non-in...

in this paper, at first we mention to some results related to pi and vertex co-pi indices and then we introduce the edge versions of co-pi indices. then, we obtain some properties about these new indices.

Every affine isometric action $\alpha$ of a group $G$ on real Hilbert space gives rise to nonsingular $\hat{\alpha}$ the associated Gaussian probability space. In recent paper [AIM19], several results ergodicity and Krieger type these actions were established when underlying orthogonal representation $\pi$ is mixing. We develop new methods prove only weakly determine in full generality. Using C...

given a graph $g$, let $g^sigma$ be an oriented graph of $g$ with the orientation $sigma$ and skew-adjacency matrix $s(g^sigma)$. then the spectrum of $s(g^sigma)$ consisting of all the eigenvalues of $s(g^sigma)$ is called the skew-spectrum of $g^sigma$, denoted by $sp(g^sigma)$. the skew energy of the oriented graph $g^sigma$, denoted by $mathcal{e}_s(g^sigma)$, is defined as the sum of the n...

Some partition theorems similar to the Rogers-Ramanujan theorems are proved. We shall prove the following partition theorems, namely THEOREM 1. The number of partitions of k, k = a 1 + a 2 + a 3 + · · · with a 1 > a 2 ≥ a 3 > a 4 ≥ · · ·

let $alpha$ be an automorphism of a ring $r$. the authors [on skewinverse laurent-serieswise armendariz rings, comm. algebra 40(1)(2012) 138-156] applied the concept of armendariz rings to inverseskew laurent series rings and introduced skew inverselaurent-serieswise armendariz rings. in this article, we study on aspecial type of these rings and introduce strongly armendariz ringsof inverse ske...

in this paper, we present some inequalities for the co-pi index involving the some topological indices, the number of vertices and edges, and the maximum degree. after that, we give a result for trees. in addition, we give some inequalities for the largest eigenvalue of the co-pi matrix of g.

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...