The extremal graphs of order trees and their topological indices

A Relation on Trees and the Topological Indices Based on Subgraph

“…T and T ′ are said to be (ϕ, µ)similar with respect to e 1 ∈ E(T ) if µ T (e) = µ T ′ (ϕ(e)) for any e e 1 . We start with a lemma which is given in [20].…”

“…They are Wiener index [25], modified Wiener indices [10], variable Wiener indices [24] and Steiner k-Wiener index [16]. Recently, Song and Huang et al [20] extended the above conclusions to the more topological indices: hyper-Wiener index [17], Wiener-Hosoya index [18], degree distance [13], Gutman index [9] and second atom-bond connectivity index [7]. All these topological indices and their edge contribution functions are summarized in the Table 1.…”

“…Instead of dealing with the extremal problem for individual topological index one by one, we can use the concept of edge additive eccentric topological index to unify the problem of determining the extreme topological index in certain classes of trees. In the literature [20] the authors gave and summarized all the results for the extremal topological indices involved in Table 1. Here, at the last of the paper, we turn to consider whether trees with different structures can have the same topological indices?…”

“…They also gave the relationship between edge division vector and the topological index on the class of trees, which enables us to simply calculate the topological indices of trees by their edge division vectors. In [20], Song and Huang et al gave a new criterion to determine the order of trees with respect to the edge division vectors. Based on these results a large of extremal trees are determined including old and new, one can refer to Table 3 in [20].…”

“…In [20], Song and Huang et al gave a new criterion to determine the order of trees with respect to the edge division vectors. Based on these results a large of extremal trees are determined including old and new, one can refer to Table 3 in [20].…”