Expected value for the <i>k</i>-distance degree index of a graph

For a graph G , the k -distance degree index is N κ (G) = ∑ κ =1 diam(G) ( ∑ w∈V(G) d κ (w) ) κ . In this article, we calculate the accurate formula of the expected value for k -distance degree index in a random arranged polygonal string with m -arranged polygons. Finally, we determine the average value of this index in the collection of all those arranged polygonal strings.