Research Interests:
In this paper, we associate the graph ΓI(N) to an ideal I of a near-ring N. We exhibit some properties and structure of ΓI(N). For a commutative ring R, Beck conjectured that both chromatic number and clique number of the zero-divisor... more
In this paper, we associate the graph ΓI(N) to an ideal I of a near-ring N. We exhibit some properties and structure of ΓI(N). For a commutative ring R, Beck conjectured that both chromatic number and clique number of the zero-divisor graph Γ(R) of R are equal. We prove that Beck's conjecture is true for ΓI(N). Moreover, we characterize all right permutable near-rings N for which the graph ΓI(N) is finitely colorable.
Research Interests:
Research Interests:
Let $L$ be a lattice with the least element $0$. An element $xin L$ is a zero divisor if $xwedge y=0$ for some $yin L^*=Lsetminus left{0right}$. The set of all zero divisors is denoted by $Z(L)$. We associate a simple graph... more
Let $L$ be a lattice with the least element $0$. An element $xin L$ is a zero divisor if $xwedge y=0$ for some $yin L^*=Lsetminus left{0right}$. The set of all zero divisors is denoted by $Z(L)$. We associate a simple graph $Gamma(L)$ to $L$ with vertex set $Z(L)^*=Z(L)setminus left{0right}$, the set of non-zero zero divisors of $L$ and distinct $x,yin Z(L)^*$ are adjacent if and only if $xwedge y=0$. In this paper, we obtain certain properties and diameter and girth of the zero divisor graph $Gamma(L)$. Also we find a dominating set and the domination number of the zero divisor graph $Gamma(L)$.