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

Faten H. Damag; Amin Saif; Adem Kiliçman; Fozaiyah Alhubairah; Khaled M. Saad; Ekram E. Ali; Mouataz Billah Mesmouli; Faten H. Damag; Amin Saif; Adem Kiliçman; Fozaiyah Alhubairah; Khaled M. Saad; Ekram E. Ali; Mouataz Billah Mesmouli

doi:10.3934/math.2025787

AIMS Mathematics

2025, Volume 10, Issue 8: 17623-17641. doi: 10.3934/math.2025787

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Research article

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

1.
Department of Mathematics, Faculty of Sciences, University of Ha'il, Ha'il 2440, Saudi Arabia
2.
Department of Mathematics, College of Sciences and Arts, Najran University, Najran, Saudi Arabia
3.
Department of Mathematics, Faculty of Applied Sciences, Taiz University, Taiz 6803, Yemen
4.
School of Mathematical Sciences, College of Computing, Informatics and Mathematics, Universiti Teknologi MARA, Shah Alam 40450, Selangor, Malaysia

Received: 22 May 2025 Revised: 19 July 2025 Accepted: 24 July 2025 Published: 05 August 2025
MSC : 05C99, 18F60, 05C20

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.
- directed graph,
- connectedness,
- continuous function,
- topology,
- path
Citation: Faten H. Damag, Amin Saif, Adem Kiliçman, Fozaiyah Alhubairah, Khaled M. Saad, Ekram E. Ali, Mouataz Billah Mesmouli. Monophonic sets and rough directed topological spaces: Applications with some directed networks[J]. AIMS Mathematics, 2025, 10(8): 17623-17641. doi: 10.3934/math.2025787

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Abstract

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.

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