On the power graphs which are Cayley graphs of some groups

In 2013, Jemal Abawajy, Andrei Kelarev and Morshed Chowdhury [1] proposed a problem to characterize the finite groups whose power graphs are Cayley graphs of some groups. Here we give a complete answer to this question

On the spectrum of linear dependence graph of finite dimensional vector spaces

In this paper, we introduce a graph structure called linear dependence graph of a finite dimensional vector space over a finite field. Some basic properties of the graph like connectedness, completeness, planarity, clique number, chromatic number etc. have been studied. It is shown that two vector spaces are isomorphic if and only if their corresponding linear dependence graphs are isomorphic. Also adjacency spectrum, Laplacian spectrum and distance spectrum of the linear dependence graph have been studied.Comment: 3 figure

On band orthorings

A semiring $S$ which is a union of rings is called completely regular, if moreover, it is orthodox then $S$ is called an orthoring. Here we study the orthorings $S$ such that $E^+(S)$ is a band semiring. Every band semiring is a spined product of a left band semiring and a right band semiring with respect to a distributive lattice. A similar spined product decomposition for the band orthorings have been proved. The interval $[\mathbf{Ri}, \mathbf{BOR}]$ is lattice isomorphic to the lattice $\mathcal{L}(\mathbf{BI})$ of all varieties of band semirings, where $\mathbf{Ri}$ and $\mathbf{BOR}$ are the varieties of all rings and band orthorings, respectively

On the prime spectrum of an le-module

Here we continue to characterize a recently introduced notion, le-modules $_{R}M$ over a commutative ring $R$ with unity \cite{Bhuniya}. This article introduces and characterizes Zariski topology on the set $Spec(M)$ of all prime submodule elements of $M$. Thus we extend many results on Zariski topology for modules over a ring to le-modules. The topological space Spec(M) is connected if and only if $R/Ann(M)$ contains no idempotents other than $\overline{0}$ and $\overline{1}$. Open sets in the Zariski topology for the quotient ring $R/Ann(M)$ induces a base of quasi-compact open sets for the Zariski-topology on Spec(M). Every irreducible closed subset of Spec(M) has a generic point. Besides, we prove a number of different equivalent characterizations for Spec(M) to be spectral.Comment: 21 page

Uniqueness of primary decompositions in Laskerian le-modules

Here we introduce and characterize a new class of le-modules $_{R}M$ where $R$ is a commutative ring with $1$ and $(M,+,\leqslant,e)$ is a lattice ordered semigroup with the greatest element $e$. Several notions are defined and uniqueness theorems for primary decompositions of a submodule element in a Laskerian le-module are established.Comment: 17 page

Rank preservers of matrices over additively idempotent and multiplicatively cancellative semirings

Here we characterize the linear operators that preserve rank of matrices over additively idempotent and multiplicatively cancellative semirings. The main results in this article generalize the corresponding results on the two element Boolean algebra and on the max algebra; and holds on max-plus algebra and some other tropical semirings.Comment: 15 page

On some characterizations of strong power graphs of finite groups

Let $G$ be a finite group of order $n$. The strong power graph $\mathcal{P}_s(G)$ of $G$ is the undirected graph whose vertices are the elements of $G$ such that two distinct vertices $a$ and $b$ are adjacent if $a^{{m}_1}$=$b^{{m}_2}$ for some positive integers ${m}_1 ,{m}_2 < n$. In this article we classify all groups $G$ for which $\mathcal{P}_s(G)$ is line graph and Caley graph. Spectrum and permanent of the Laplacian matrix of the strong power graph $\mathcal{P}_s(G)$ are found for any finite group $G$.Comment: 13 page

On some properties of enhanced power graph

Given a group $G$, the enhanced power graph of $G$ denoted by $\mathcal{G}_e(G)$, is the graph with vertex set $G$ and two distinct vertices $x, y$ are edge connected in $\mathcal{G}_e(G)$ if there exists $z\in G$ such that $x=z^m$ and $y=z^n$, for some $m, n\in \mathbb{N}$. In this article, we characterize the enhanced power graph $\mathcal{G}_e(G)$ of $G$. The graph $\mathcal{G}_e(G)$ is complete if and only if $G$ is cyclic, and $\mathcal{G}_e(G)$ is Eulerian if and only if $|G|$ is odd. We classify all abelian groups and also all non-abelian $p-$groups $G$ for which $\mathcal{G}_e(G)$ satisfies the cone property.Comment: 10 page

Normal Subgroup Based Power Graph of a finite Group

For a finite group $G$ with a normal subgroup $H$, the normal subgroup based power graph of $G$, denoted by $\Gamma_H(G)$ whose vertex set $V(\Gamma_H(G))=(G\setminus H)\bigcup \{e\}$ and two vertices $a$ and $b$ are edge connected if $aH=b^mH$ or $bH=a^nH$ for some $m, n \in \mathbb{N}$. In this paper we obtain some fundamental characterizations of the normal subgroup based power graph. We show some relation between the graph $\Gamma_H(G)$ and the power graph $\Gamma(\frac{G}{H})$. We show that $\Gamma_H(G)$ is complete if and only of $\frac{G}{H}$ is cyclic group of order $1$ or $p^m$, where $p$ is prime number and $m\in \mathbb{N}$. $\Gamma_H(G)$ is planar if and only if $|H|=2$ or $3$ and $\frac{G}{H}\cong \mathbb{Z}_2\times \mathbb{Z}_2 \times \cdots \times \mathbb{Z}_2$. Also $\Gamma_H(G)$ is Eulerian if and only if $|G|\equiv |H|$ mod$2$.Comment: 14 pages, 4 figure

An elementary proof of a conjecture on graph-automorphism

In this article, we give an elementary combinatorial proof of a conjecture about the determination of automorphism group of the power graph of finite cyclic groups, proposed by Doostabadi, Erfanian and Jafarzadeh in 2013.Comment: 5 page