On k-Distance Degree Index of Trees

Let G be a connected graph then ܰ௞-index (݇-distance degree index) defined in [14] as ܰ௞(ܩ)=∑ௗ௜௔௠(ீ)௞ୀଵቀ∑௫ఢ௏(ீ)݀௞(ݔ)ቁ݇, where ݀௞(ݔ)=|ܰ௞(ݔ)|=|{ݕܸ߳(ܩ):݀(ݔ,ݕ)=݇}|, where ݀(ݔ,ݕ)is the distance between vertex ݔand ݕin graph ܩand ݉ܽ݅݀(ܩ)is the diameter of graph ܩ. We define some transformations and their impact on ܰ௞-index of graphs with respect to pendant path and pendant vertices. For fixed number of pendant vertices of a tree, we define a tree with minimum ܰ௞-index. Also for different fixed parameters we characterize the tress with minimum ܰ௞-index