On total directed graphs of non-commutative rings

Abstract For a non-commutative ring R , the left total directed graph of R is a directed graph with vertex set as R and for the vertices x and y , x is adjacent to y if and only if there is a non-zero r ∈ R which is different from x and y , such that r x + y r is a left zero-divisor of R . In this paper, we discuss some very basic results of left (as well as right) total directed graph of R . We also study the coloring of left total directed graph of R directed graph.