arXiv subjects

Anita Lande

Publications and source records attributed to Anita Lande.

3 recordsLinked to original sources

arXiv2024

Generalized zero-divisor graph of $*$-rings

Let $R$ be a ring with involution $*$ and $Z^*(R)$ denotes the set of all non-zero zero-divisors of $R$. We associate a simple (undirected) graph $Γ'(R)$ with vertex set $Z^*(R)$ and two distinct vertices $x$ and $y$ are adjacent in $Γ'(R)$ if and only if $x^ny^*=0$ or $y^nx^*=0$, for some positive integer $n$. We find the diameter and girth of $Γ'(R)$. The characterizations are obtained for $*$-rings having $Γ'(R)$ a connected graph, a complete graph, and a star graph. Further, we have shown that for a ring $R$, there is an involution on $R\times R$ such that $Γ'(R\times R)$ is disconnected if and only if $R$ is an integral domain.

math.CO↗

arXiv2023

On spectrum of the zero-divisor graph of matrix ring

For a ring $R$, the zero-divisor graph is a simple graph $Γ(R)$ whose vertex set is the set of all non-zero zero-divisors in a ring $R$, and two distinct vertices $x$ and $y$ are adjacent if and only if $xy=0$ or $yx=0$ in $R$. By using Weyl's inequality we give bounds on eigenvalues of adjacency matrix of $Γ(M_2(F))$, where $M_2(F)$ is a $2 \times 2$ matrix ring over a finite field $F$.

math.SP↗

arXiv2022

Spectra of the zero-divisor graph of finite rings

The zero-divisor graph $Γ(R)$ of a ring $R$ is a graph with nonzero zero-divisors of $R$ as vertices and distinct vertices $x,y$ are adjacent if $xy=0$ or $yx=0$. We provide an equivalence relation on a ring $R$ and express $Γ(R)$ as a generalized join of graphs on equivalence classes of this relation. We determined the adjacency and Lapalcian spectra of $Γ(R)$ when $R$ is a finite semisimple ring.

math.SP↗