On the local (a, d)-edge antimagic coloring of gear graph and its subdivision

For any graph G = (V, E), with |V(G)| = p and |E(G)|= q. A bijection f : E(G) → {1, 2, 3,...,|E(G)|} is called an local (a,d)-edge antimagic coloring of G if the element of the edge-weight set w(uv) = f(u) + f(v), where w(e1) ≠ w(e2), uv ∈ E(G), are distinct and the set of all edge-weights are formed an arithmetic sequence with initial smallest edge-weight value a and different d. The local (a,d)-edge antimagic chromatic number is is the minimum number of colors needed to color G such that a graph G admits the local (a,d)-edge antimagic coloring.The local (a,d)-edge antimagic chromatic number denoted by χ’(a,d)-ela. In the present, we will obtain the lower and upper bound of χ’(a,d)-ela and determine the exact of value of the local (a,d)-edge antimagic coloring chromatic number of gear graph and it’s subdivisions.