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Let R be a commutative ring and M be a unital R-module. The intersection graph of submodules of M, denoted by GM, is the graph whose vertex set is the collection of all submodules of M and in which two distinct vertices A and B are adjacent if and only if A∩B≠0. In this paper the notion of essentiality of modules plays a vital role in the study of intersection graph of submodules of M. This notion gives a new dimension in characterizing the center of intersection graphs of submodules of M. We define mna (maximal non-adjacent) vertex in GMand observe some of its characteristics. The notion of complemented intersection graph exhibits some significant algebraic and graphical properties. Moreover, defining the concept of isolated center in GMwe establish certain results related to mna vertex. Keywords: Module, Essential submodule, Intersection graph, Complemented graph, Center

Let be a commutative ring and be a unital -module. The intersection graph of submodules of , denoted by , is the graph whose vertex set is the collection of all submodules of and in which two distinct vertices and are adjacent if and only if . In this paper the notion of essentiality of modules plays a vital role in the study of intersection graph of submodules of . This notion gives a new dimension in characterizing the center of intersection graphs of submodules of . We define mna (maximal non-adjacent) vertex in and observe some of its characteristics. The notion of complemented intersection graph exhibits some significant algebraic and graphical properties. Moreover, defining the concept of isolated center in we establish certain results related to mna vertex.

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