On the Study of Rainbow Antimagic Connection Number of Comb Product of Friendship Graph and Tree

Given a graph G with vertex set (Formula presented.) and edge set (Formula presented.), for the bijective function (Formula presented.), the associated weight of an edge (Formula presented.) under f is (Formula presented.). If all edges have pairwise distinct weights, the function f is called an edge-antimagic vertex labeling. A path P in the vertex-labeled graph G is said to be a rainbow (Formula presented.) path if for every two edges (Formula presented.) it satisfies (Formula presented.). The function f is called a rainbow antimagic labeling of G if there exists a rainbow (Formula presented.) path for every two vertices (Formula presented.). We say that graph G admits a rainbow antimagic coloring when we assign each edge (Formula presented.) with the color of the edge weight (Formula presented.). The smallest number of colors induced from all edge weights of antimagic labeling is the rainbow antimagic connection number of G, denoted by (Formula presented.). This paper is intended to investigate non-symmetrical phenomena in the comb product of graphs by considering antimagic labeling and optimizing rainbow connection, called rainbow antimagic coloring. In this paper, we show the exact value of the rainbow antimagic connection number of the comb product of graph (Formula presented.), where (Formula presented.) is a friendship graph with order (Formula presented.) and (Formula presented.), where (Formula presented.) is the path graph of order m, (Formula presented.) is the star graph of order (Formula presented.), (Formula presented.) is the broom graph of order (Formula presented.) and (Formula presented.) is the double star graph of order (Formula presented.).