Topological Approaches of Graphs using $j$-neighbourhoods and Their Applications (European Journal of Pure and Applied Mathematics, 2025)
This paper investigates novel topological structures on graphs through the lens of $j$-neighbourhoods, specifically out, in, intersection, and union-based neighbourhoods. We develop a systematic framework for constructing subbases and topologies on directed graphs using these neighbourhoods and analyze their topological properties. Our work provides a rigorous comparative study of neighbourhood types, their interrelations, and their role in generating induced topologies. In addition, we explore potential applications in digital topology, spatial networks, and data structure. The theoretical results are supported by aircraft paths on an airline as an illustrative example and comparison tables that highlight structural differences and practical implications.
Keywords: Directed graphs; Topological spaces; j-neighbourhoods; In-neighbourhood; Out-neighbourhood; Graph topology; Subbase construction