The Harary index of ordinary and generalized quasi-tree graphs

• Published in: Journal of applied mathematics & computing Vol. 45; no. 1-2; pp. 365 - 374
• Main Authors: Xu, Kexiang, Wang, Jinlan, Liu, Hongshuang
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.06.2014; Springer Nature B.V
• Subjects: Applied mathematics; Computation; Computational Mathematics and Numerical Analysis; Equality; Graph theory; Graphs; Lower bounds; Mathematical analysis; Mathematical and Computational Engineering; Mathematical models; Mathematics; Mathematics and Statistics; Mathematics of Computing; Original Research; Studies; Theory of Computation; Trees; 05C90; 05C12; Quasi-tree graph; Distance (in graph); 05C05; Harary index
• ISSN: 1598-5865; 1865-2085
• DOI: 10.1007/s12190-013-0727-4

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 V k ⊆ V ( G ) where | V k |= k such that G − V k 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.