On the ordering of distance-based invariants of graphs.

Let d ( u, v ) be the distance between u and v of graph G , and let W f ( 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, W f ( 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 n  ≥ 22 (resp. n  ≥ 38) and the ten smallest hyper-Wiener indices of trees of order n  ≥ 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). [ABSTRACT FROM AUTHOR]