A Study on b-chromatic number of comb product of cycle graph related graphs

The b-chromatic number of a graph, written as φ(G), is the highest number of colors you can use to color the graph properly, with one special rule: in each color group, there must be at least one vertex that’s connected to vertices of all other color groups. This kind of coloring is called b-coloring.In this study, we look at two connected graphs G and H.A special graph operation called the comb product is used to combine them. This operation, denoted by G⊳H, works like this: take one copy of G, and then take as many copies of H as there are vertices in G. Then, attach each copy of H to a vertex in G at a specific point called vertex o in H.This research explores how b-coloring works when we apply this comb product to different graphs, especially those related to cycle graphs. The goal is to understand how the structure of the combined graph affects its b-chromatic number.