NILPOTENT GRAPHS OF MATRIX ALGEBRAS

Let R be a ring with unity. The undirected nilpotent graph of R, denoted by ΓN(R), is a graph with vertex set ~ZN(R)∗={0≠x∈R| xy∈N(R) for some y∈R∗}, and two distinct vertices x and y are adjacent if and only if xy∈N(R), or equivalently, yx∈N(R), where N(R) denoted the nilpotent elements of R. Recently, it has been proved that if R is a left Artinian ring, then \diam(ΓN(R))⩽3. In this paper, using the concept of rank over commutative rings, we investigate basic properties of undirected nilpotent graph of matrix algebras. Moreover, some result on undirected nilpotent graph of matrix algebras over commutative rings are given. For instance, we prove that ΓN(Mn(R)) is not planar for all n⩾2. Furthermore, we show that \diam(ΓN(R))⩽\diam(ΓN(Mn(R))) for an Artinian commutative ring R. Also, we prove that ΓN(Mn(R))≅ΓN(Mn(T(R))), where T(R) be the total quotient ring of a commutative ring R