Suppose that $A_1$ and $A_2$ are two linear codes over $F_m$ with parameters $[n,k]$ and $[n,k^{\prime }]$, respectively. For the code $A=\left\{\right(x,x+y):x\in A_1,y\in A_2\}$, we asked about $d_r\left(A\right)$ in terms of $d_r\left(A_1\right)$ and $d_r\left(A_2\right)$, so we obtained an upper bound for $d_r\left(A\right)$ in terms of them. In addition, we obtained $d_2\left(A\right)$ in which $A_1$ and $A_2$ are codes over $F_2$. Additionally, we obtained $d_2\left(A\right)$ in non-binary case, where it is different of what we have for binary case. Suppose that $M$ is a linear code over a finite field. The minimum of the support sizes among $r$-dimensional sub-codes of $M$ is called the $r$-th generalized Hamming weight of $M$ that we will denote it by $d_r\left(M\right)$.

Its well known that there are a limited number of method which associate group theory to graph theory. In this paper, we propose a new method and obtain a novel kind of graphs. Some important application like graph energy was established to the proposed graph.

Let $n$ be a positive integer. The intersection graph of ring $R={\mathrm{\mathbb{Z}}}_n$, is a graph whose vertices are non-empty and proper ideals of ${\mathrm{\mathbb{Z}}}_n$ and two vertices are adjacent if and only if corresponding ideals have non-zero intersection and is denoted by $G\left({\mathrm{\mathbb{Z}}}_n\right)$. In this paper we compute some topological indices of this graph.

Let $\left(\mathrm{\Gamma }\right(R){)}^c$ be the maximal graph of a commutative ring with identity $R$ whose vertices are elements of $R$. Two distinct vertices $a$ and $b$ are adjacent if and only if $Ra+Rb\ne R$ or there exists a maximal ideal of $R$ containing both them. In this paper, we study some properties of two (induced) subgraphs $\left({\mathrm{\Gamma }}_2\right(R){)}^c$ and $\left(G\right(R){)}^c$ of $\left(\mathrm{\Gamma }\right(R){)}^c$.

The concepts of $2$-crossed polymodule of polygroups, and pullback and pushout $2$-crossed polymodules are introduced and some of their basic properties are investigated. In addition, the construction of pullback and pushout $2$-crossed polymodules are described.

A vertex subset $W\subseteq V$ of the graph $G=(V,E)$ is an independent dominating set, if every vertex in $V\setminus W$ is adjacent to at least one vertex in $W$ and the vertices of $W$ are pairwise non-adjacent. We enumerate independent dominating sets in several classes of graphs (polymer graph) made by a linear or cyclic concatenation of basic building blocks. Explicit recurrences are derived for the number of independent dominating sets of these kind of graphs. Generating functions for the number of independent dominating sets of triangular and squares cacti chain are also computed.

Let $G=(V,E)$ be a simple graph. A dominating set of $G$ is a subset $D\subseteq V$ such that every vertex not in $D$ is adjacent to at least one vertex in $D$. The cardinality of a smallest dominating set of $G$, denoted by $\gamma \left(G\right)$, is the domination number of $G$. For $k\ge 1$, a $k$-fair dominating set ($kFD$-set) in $G$, is a dominating set $D$ such that $\left|N\right(v)\cap D|=k$ for every vertex $v\in V\setminus D$. A fair dominating set, in $G$ is a $kFD$-set for some integer $k\ge 1$. In this paper, after presenting preliminaries, we count the number of fair dominating sets of some specific graphs.

For a finite group G, an irreducible character of $G$ is called a $\mathcal{P}$-character, if it is a constituent of permutation character $(1_M{)}^G$ for some maximal subgroup $M$ of $G$. In this article, we determine the $\mathcal{P}$-characters of group $PSL(2,q)$, and we check which of the permutation characters $(1_M{)}^{PSL(2,q)}$ are multiplicity-free.

Finding coefficients of Laplacian and signless Laplacian characteristic polynomial of graphs is an algebraic graph theory problem that scientists have sought to answer it for these years. The purpose of this article is finding relationships between coefficients of Laplacian and signless Laplacian characteristic polynomial of polythiophenes.