Hamiltonian trace graph of matrices.

Let R be a commutative ring with identity, n ≥ 2 be a positive integer and M n (R) be the set of all n × n matrices over R. For a matrix A ∈ M n (R) , Tr (A) is the trace of A. The trace graph of the matrix ring M n (R) , denoted by Γ t (M n (R)) , is the simple undirected graph with vertex set { A ∈ M n (R) ∗ : there exists  B ∈ M n (R) ∗  such that Tr (A B) = 0 } and two distinct vertices A and B are adjacent if and only if Tr (A B) = 0. The ideal-based trace graph of the matrix ring M n (R) with respect to an ideal I of R , denoted by Γ I t (M n (R)) , is the simple undirected graph with vertex set M n (R) ∖ M n (I) and two distinct vertices A and B are adjacent if and only if Tr (A B) ∈ I. In this paper, we investigate some properties and structure of Γ I t (M n (R)). Further, it is proved that both Γ t (M n (R)) and Γ I t (M n (R)) are Hamiltonian. [ABSTRACT FROM AUTHOR]