Multiplicatively weighted Harary index of graphs

摘要

Let be a connected graph with vertex set . The multiplicatively weighted Harary index of a graph is defined as , where is the degree of vertex , and denotes the distance between and . In this paper, we first prove that the multiplicatively weighted Harary index of a graph is monotonic on some transformations, and then determine the extremal values of the multiplicatively weighted Harary indices for some familiar classes of graphs and characterize the corresponding extremal graphs.