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

Given a graph G with vertex set V (G) and edge set E (G) , for the bijective function f (V (G)) → { 1 , 2 , ⋯ , | V (G) | } , the associated weight of an edge x y ∈ E (G) under f is w (x y) = f (x) + f (y) . 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 x − y path if for every two edges x y , x ′ y ′ ∈ E (P) it satisfies w (x y) ≠ w (x ′ y ′) . The function f is called a rainbow antimagic labeling of G if there exists a rainbow x − y path for every two vertices x , y ∈ V (G) . We say that graph G admits a rainbow antimagic coloring when we assign each edge x y with the color of the edge weight w (x y) . The smallest number of colors induced from all edge weights of antimagic labeling is the rainbow antimagic connection number of G, denoted by r a c (G) . 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 F n ⊳ T m , where F n is a friendship graph with order 2 n + 1 and T m ∈ { P m , S m , B r m , p , S m , m } , where P m is the path graph of order m, S m is the star graph of order m + 1 , B r m , p is the broom graph of order m + p and S m , m is the double star graph of order 2 m + 2 . [ABSTRACT FROM AUTHOR]