The Harary Index Of A Unicyclic Graph

LetG = (V, E) be a simple and connected graph with vertex set V and edge set E. The Harary index of G is denned aswhere d(u, v) is the length of the shortest path between two distinct vertices u and v.Harary index is a topological index introduced independently by Plavsic and Ivanciuc et al. in 1993. combining with the related indices appears to be very efficacious in improving the QSPR models.In this paper, we first give a formula for calculating the Harary index of a unicyclic graph according to the structure, and then using this formula, we give the largest, the second largest and the third largest Harary index of unicyclic graphs, respectively. Also, we characterize the extremal graphs with respect to the Harary index.