The trace graph of the matrix ring over a finite commutative ring

Ideal based trace graph of matrices

“…, C i|Mn(I)| to the vertices of Γ I t (M n (R)) arising out of the vertices of the color class C i we have k|M n (I)| colors and the coloring is proper. Thus χ(Γ I t (M n (R))) ≤ k|M n (I)|.The following theorem is a generalization of the moreover case of Theorem 4.1(1). Let n ≥ 2 be an integer, R be a commutative ring and I be a non trivial ideal in R. Let S be a clique of maximum order in Γ t (M n (R/I)) and S have the largest number of elements A with Tr(A 2 ) ∈ I.…”

“…Later on, Anderson and Livingston [2] modified the definition with vertex set, the set of all nonzero zero divisors of R and introduced the zero-divisor graph Γ(R) corresponding to a commutative ring R. In [9], Redmond introduced the notion of the zero-divisor graph with respect to an ideal I of a commutative ring R, denoted by Γ I (R), as the graph with vertex set {x ∈ R \ I : xy ∈ I for some y ∈ R \ I}, and two distinct vertices x and y are adjacent if and only if xy ∈ I. The concept of trace graph of a matrix ring over a commutative ring was introduced by Almahdi, Louartiti, and Tamekkante [1]. Several authors have extensively studied about zero-divisor graph with respectxzs to an ideal.…”

Induced subgraphs of zero-divisor graphs

“…Another very general problem is to examine the induced subgraphs of various generalizations of zero-divisor graphs, such as the extended zerodivisor graphs and the trace graphs [2,24,25,26,27].…”