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 + yr 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.