Arbitrarily large difference between -strong chromatic index and its trivial lower bound

Let φ : E → { 1 , 2 , … , k } be a proper edge coloring of a graph G = ( V , E ) . The set of colors of edges incident to a vertex v ∈ V is called the color set of v and denoted by S ( v ) . The coloring φ is vertex-distinguishing if S ( u ) ≠ S ( v ) for any two distinct vertices u , v ∈ V . trong edge coloring of a graph G is a proper edge coloring that distinguishes any two distinct vertices u and v with distance 1 ≤ d ( u , v ) ≤ d . The minimum number of colors of a d -strong edge coloring of graph G is called d -strong chromatic index of G and denoted by χ d ′ ( G ) . sent some general results on d -strong edge colorings of circulant graphs. We determine exact values of the d -strong chromatic index for circulant graphs C n ( 1 , 2 ) for d = 1 and d = 2 , we also prove that the difference between the d -strong chromatic index and its trivial lower bound can be arbitrarily large.