The Harary index of ordinary and generalized quasi-tree graphs

Abstract

The Harary index is defined as the sum of reciprocals of distances between all pairs of vertices of a connected graph. The quasi-tree graph is a graph G in which there exists a vertex v ∈ V( G) such that G − v is a tree. In this paper, we presented the upper and lower bounds on the Harary index of all quasi-tree graphs of order n and characterized the corresponding extremal graphs. Moreover we defined the k- generalized quasi-tree graph to be a connected graph G with a subset Vk ⊆ V( G) where |Vk |= k such that G − Vk is a tree. And we also determined the k-generalized quasi-tree graph of order n with maximal Harary index for all values of k and the extremal one with minimal Harary index for k = 2.