Monophonic sets and rough directed topological spaces: Applications with some directed networks

By using the monophonic paths in the theory of directed graphs, this paper constructs a new topology, called the out mondirected topology, and characterizes the graphs that induce the indiscrete or discrete topology. We give and study some relations and properties such as the relationship between the isomorphic relation in directed graphs and the homeomorphic property in out mondirected topological spaces, compactness, ${\mathbb{D}}_{\pm }$-connectedness, connectedness, and ${\mathbb{D}}_{\pm }$-discrete properties. Finally, we apply our results of out mondirected topological spaces in the nervous system of the human body, such as in the messenger signal network, in diagrams of sensory neuron cells, and in models of two distinct nicotinic receptor types based on second messenger signal.