On the modified misbalance rodeg index

Let G be a graph with edge set E(G).Denote by du the degree of a vertex u in G.The number vw∈E(G) √ dv -√ dw 2 is known as the modified misbalance rodeg (MMR) index of G.The primary goal of this paper is to characterize graphs attaining the maximum and minimum values of the MMR index in the classes of all trees and connected unicyclic graphs with a fixed order.In the case of trees, the path and star graphs attain the minimum and maximum values of the MMR index, respectively.For the case of unicyclic graphs, the cycle graph and the graph formed by inserting an edge in the star graph achieve the minimum and maximum values of the MMR index, respectively.