Study on Special Bipartite Graph through Topological Framework

A unique class of bipartite graphs named "Multi-K<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2,2</inf> Ladder Graph" formed by linking multiple K<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{2,2}</inf> graphs in sequence, resulting in a structure that resembles and extends the traditional ladder graph has been presented. These graphs have a characteristic degree distribution, where the end vertices have fewer connections than the ones in the middle. We explore how minimal adjacency relationships among the vertices can be used to generate a topological space. Using this topological framework, we explore the concept of order of each vertex related to the induced adjacency topology. This perspective creates a bridge between graph theory and topology, opening up new possibilities for analyzing structured data and networks.