A submodule, Φ , of ℤ_4^n is called a double cyclic code of length n=k+l over ℤ_4 if any cyclic shift of the first k coordinates and last l coordinates of a codeword is a codeword. The code Φ is called a separable code, if Φ =Φ _k ×Φ _l where Φ _k is the canonical projection of Φ on the first k coordinates and Φ _l on the last l coordinates. If there exists a basis for Φ , then Φ is called free. Also, the code Φ is a self-dual code, if Φ =Φ ^⊥ . The self-dual code Φ is Type II if the Euclidean weight of every codeword is divisible by 8, otherwise is Type I. In this paper, we study free double cyclic self-dual codes of length n=k+l over ℤ_4 . We investigate the separable free double cyclic codes of length n=k+l over ℤ_4 and show that the separable free double cyclic codes are not self-dual codes. Moreover, we provide the process of finding the nonseparable free double cyclic self-dual codes of length n=k+l over ℤ_4 . In particular, we determine n=8 is the shortest length of the codes. Finally, we classify Type I and Type II codes of the nonseparable self-dual codes and study optimal codes of them.

Dendrimer molecules are macromolecules which have many applications in nanosciences, drug delivery, biology, and different areas of sciences. Topological indices of chemical graph theory are numerical descriptor of a molecular structure. The dendrimer graph Gn is obtained by attaching the new paths P9, joined each pendant vertex of Gn−1 to central vertex of P9. Also, the dendrimer graph Hn is obtained by attaching the new paths P15, joined each pendant vertex of Hn−1 to central vertex of P15. In this paper, we study topological indices of dendrimer graphs Gn and Hn. Also, we obtain the Szeged index, Wiener index, Steiner k–Wiener index, Schultz index, Gutman index, Padmakar–Ivan index, first Zagreb index, and second Zagreb index of dendrimer graphs Gn and Hn.

Dendrimer molecules are macromolecules which have many applications in nanosciences, drug delivery, biology, and different areas of sciences. Topological indices of chemical graph theory are numerical descriptor of a molecular structure. The dendrimer graph G[n] is obtained by attaching the new paths P9, joined each pendant vertex of G[n − 1] to central vertex of P9. Also, the dendrimer graph H[n] is obtained by attaching the new paths P15, joined each pendant vertex of H[n − 1] to central vertex of P15. In this paper, we study topological indices of dendrimer graphs G[n] and H[n]. Also, we obtain the Szeged index, Wiener index, Steiner k–Wiener index, Schultz index, Gutman index, Padmakar–Ivan index, first Zagreb index, and second Zagreb index of dendrimer graphs G[n] and H[n].

A Boolean function is a function $f:\Bbb{Z}_n^2 \rightarrow \{0,1\}$ and we denote the set of all $n$-variable Boolean functions by $BF_n$. For $f\in BF_n$ the vector $[{\rm W}_f(a_0),\ldots,{\rm W}_f(a_{2n-1})]$ is called the Walsh spectrum of $f$, where ${\rm W}_f(a)= \sum_{x\in V} (-1)^{f(x) \oplus ax}$, where $V_n$ is the vector space of dimension $n$ over the two-element field $F_2$. In this paper, we shall consider the Cayley graph $\Gamma_f$ associated with a Boolean function $f$. We shall also find a complete characterization of the bent Boolean functions of order $16$ and determine the spectrum of related Cayley graphs.In addition, we shall enumerate all orbits of the action of automorphism group on the set $BF_n$.

In the present paper we study some properties of a new graph invariant named reciprocal degree distance of some molecular graphs. This new topological index is defined by Hua et al. as RDD(G) = Sigma(u,v is an element of E(G)) (d(u) + d(v))[d(u, v)](-1), where the d(u, v) denotes the distance between vertices u and v.

Recently, Hua et al. defined a new topological index based on degrees and inverse of distances between all pairs of vertices. They named this new graph invariant as reciprocal degree distance as 1 { , } ( ) ( ( ) ( ))[ ( , )] RDD(G) = u v V G d u  d v d u v , where the d(u,v) denotes the distance between vertices u and v. In this paper, we compute this topological index for Grassmann graphs.

In this paper we study the existence of commuting regular elements in the groupoid Zn(t, u). We define the notion left (right) commuting regular elements and study its properties. Also we show that Zn(t, u) contains commuting regular subsemigroup and give a necessary and sufficient condition for the groupoid Zn(t, u) to be commuting regular. Mathematics Subject Classification: 15A27, 20M16, 20L05

In this paper, we giving certain properties of the commuting regular semigroups, we get signicant results on semigroups. Our investigation involve certain inter

In this paper, we study some main properties of the commuting regular ideals and we give a necessary and sucient condition that a ring (or semigroups ) is

A number of main properties of the commuting regular rings and commuting regular semigroups have been studied in this paper. Some significant results of which will be used for the commutative rings and a necessary and sufficient condition is given for a semigroup to be commuting regular.

Two elements x and y of a ring R are commuting regular if for some a ∈ R, xy = yxayx holds.In this paper we study the finite rings Z p [S] and Z p i 1 p 2 [L n (m)], and prove that the first one is commuting regular and the second ring contains the commuting regular element and idempotents as well (where p, p 1 and p 2 are odd primes.Moreover, i, m and n are positive integers such that m < n, (m, n) = 1 and (m -1, n) = 1).