Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
8901215 | Applied Mathematics and Computation | 2018 | 11 Pages |
Abstract
Let d(u, v) be the distance between u and v of graph G, and let Wf(G) be the sum of f(d(u, v)) over all unordered pairs {u, v} of vertices of G, where f(x) is a function of x. In some literatures, Wf(G) is also called the Q-index of G. In this paper, some unified properties to Q-indices are given, and the majorization theorem is illustrated to be a good tool to deal with the ordering problem of Q-index among trees with n vertices. With the application of our new results, we determine the four largest and three smallest (resp. four smallest and three largest) Q-indices of trees with n vertices for strictly decreasing (resp. increasing) nonnegative function f(x), and we also identify the twelve largest (resp. eighteen smallest) Harary indices of trees of order nâ¯â¥â¯22 (resp. nâ¯â¥â¯38) and the ten smallest hyper-Wiener indices of trees of order nâ¯â¥â¯18, which improve the corresponding main results of Xu (2012) and Liu and Liu (2010), respectively. Furthermore, we obtain some new relations involving Wiener index, hyper-Wiener index and Harary index, which gives partial answers to some problems raised in Xu (2012).
Related Topics
Physical Sciences and Engineering
Mathematics
Applied Mathematics
Authors
Muhuo Liu, Kinkar Ch. Das,