Narrow Results

Time-varying coverage, namely sweep coverage is a recent development in the area of wireless sensor networks, where a few mobile sensors sweep or monitor a comparatively large number of locations periodically. In this article, we study barrier sweep coverage with mobile sensors where the barrier is considered as a finite length continuous curve on a plane. The coverage at every point on the curve is time-variant. We propose an optimal solution for sweep coverage of a finite length continuous curve. Usually, energy source of a mobile sensor is a battery with limited power, so energy restricted sweep coverage is a challenging problem for long running applications. We propose an energy-restricted sweep coverage problem where every mobile sensor must visit an energy source frequently to recharge or replace its battery. We propose a $\frac{13}{3}$-approximation algorithm for this problem. The proposed algorithm for multiple curves achieves the best possible approximation factor 2 for a special case. We propose a 5-approximation algorithm for the general problem. As an application of the barrier sweep coverage problem for a set of line segments, we formulate a data gathering problem. In this problem a set of mobile sensors is arbitrarily monitoring the line segments one for each. A set of data mules periodically collects the monitoring data from the set of mobile sensors. We prove that finding the minimum number of data mules to collect data periodically from every mobile sensor is NP-hard and propose a 3-approximation algorithm to solve it.

The zero-divisor graph Γ(R) of an associative ring R is the graph whose vertices are all non-zero (one-sided and two-sided) zero-divisors of R, and two distinct vertices x and y are joined by an edge if and only if xy = 0 or yx = 0. [S. P. Redmond, The zero-divisor graph of a noncommutative ring, Int. J. Commut. Rings1(4) (2002) 203–211.]

In the present paper, all finite rings with Eulerian zero-divisor graphs are described.

Let R be a commutative ring. The intersection graph of gamma sets in the zero-divisor graph Γ(R) of R is the graph I_Γ(R) with vertex set as the collection of all gamma sets of the zero-divisor graph Γ(R) of R and two distinct vertices A and B are adjacent if and only if A ∩ B ≠ ∅. In this paper, we study about various properties of I_Γ(R) and investigate the interplay between the graph theoretic properties of I_Γ(R) and the ring theoretic properties of R.

Let $R$ be a commutative ring with $1$. Let $G(R)$ be a graph with vertices as elements of $R$, where two distinct vertices $a$ and $b$ are adjacent if and only if $aR+bR=eR$ for some non-zero idempotent $e$ in $R$. In this paper, we establish a relation between completeness of the graph $G(R)$ and regularity of the ring $R$. For a finite commutative ring $R$ with $1$, we show that the chromatic number of $G(R)$ is equal to the number of regular elements in $R$. Moreover, we characterize some graph theoretic properties of $G(R)$ and finally we characterize Eulerian property of the graph $G(R)$.

Let $m>1$ be an integer, and let $I{({\mathbb{Z}}_m)}^{\ast }$ be the set of all non-zero proper ideals of ${\mathbb{Z}}_m$. The intersection graph of ideals of ${\mathbb{Z}}_m$, denoted by $G({\mathbb{Z}}_m)$, is a graph with the vertex set $I{({\mathbb{Z}}_m)}^{\ast }$ and two distinct vertices $I,J\in I{({\mathbb{Z}}_m)}^{\ast }$ are adjacent if and only if $I\cap J\ne 0$. Let $n>1$ be an integer and ${\mathbb{Z}}_n$ be a ${\mathbb{Z}}_m$-module. In this paper, we study a kind of graph structure of ${\mathbb{Z}}_m$, denoted by $G_n({\mathbb{Z}}_m)$. It is the undirected graph with the vertex set $I{({\mathbb{Z}}_m)}^{\ast }$, and two distinct vertices $I$ and $J$ are adjacent if and only if $I{\mathbb{Z}}_n\cap J{\mathbb{Z}}_n\ne 0$. Clearly, $G_m({\mathbb{Z}}_m)=G({\mathbb{Z}}_m)$. Let $m=p_1^{{\alpha }_1}\cdots p_s^{{\alpha }_s}$ and $n=p_1^{{\beta }_1}\cdots p_s^{{\beta }_s}$, where $p_i$’s are distinct primes, ${\alpha }_i$’s are positive integers, ${\beta }_i$’s are non-negative integers, and $0\le {\beta }_i\le {\alpha }_i$ for $i=1,\dots ,s$ and let $S=\{1,\dots ,s\}$, $S^{\prime }=\{i\in S:{\beta }_i\ne 0\}$. The cardinality of $S^{\prime }$ is denoted by $s^{\prime }$. Also, let $\alpha (G_n({\mathbb{Z}}_m))$, $\gamma (G_n({\mathbb{Z}}_m))$ and $𝔄$ denote the independence number, the domination number and the set of all isolated vertices of $G_n({\mathbb{Z}}_m)$, respectively. We prove that $gr(G_n({\mathbb{Z}}_m))\in \{3,\infty \}$ and we show that if $G_n({\mathbb{Z}}_m)$ is not a null graph, then