On mc-hop Connectedness in the Monophonic c-topological Spaces: Applications on Some Networks in the Human Optical System

In this paper, we introduce the concept of mc-vertices in simple graphs and use monophonic paths to define a new class of vertex topologies, called monophonic c-topologies. We investigate fundamental properties of these spaces, including openness-minimizing behavior, compactness, and various forms of connectedness, and we characterize graphs that induce discrete or indiscrete monophonic c-topologies. We further examine the relationship between graph isomorphisms and homeomorphisms in monophonic c-spaces. As a main contribution, we introduce mc-hop connectedness as a new measure based on monophonic eccentricity and analyze its connections with existing graphical topologies. As an application, we study monophonic c-connectedness, mc-hop connectedness, and discreteness in network models of the human optical system, including cross-sectional structures, slide clips, and visual field representations. Our results demonstrate a strong correspondence between graphical and topological structures and highlight the effectiveness of these concepts in modeling optical and biological systems.