A note on extremal trees with degree conditions
Introduction
The study of the so-called chemical indices and the associated extremal problems has been a big part of research in chemical graph theory, where the molecular structures of a chemical compound are analyzed through various graph invariants/indices that were found to correlate to properties of these compounds. Among numerous indices introduced in recent years, we are particularly interested in the distance-based and degree-based indices. Because of many applications related to acyclic structures, extremal structures among trees appear to be of interest. Furthermore, as the vertex degrees correspond to the valences of atoms, trees under various degree conditions have been extensively studied.
After presenting the necessary background information in this section, we will first survey some recent extremal results on trees under degree conditions in Section 2. Of particular interest to us are trees of a given degree sequence. We will first introduce the basic definitions and extremal structures in this section. We will then present several main extremal results among trees of a given degree sequence and discuss their immediate consequences in Section 3. Among these consequences are many previously established extremal results as well as some new ones.
All graphs considered in this paper are finite and simple. For a graph G, we use V(G), E(G), |V(G)|, e(G), Δ(G), β, α, γ to denote the vertex set, edge set, number of vertices, number of edges, maximum degree, matching number, independent number, domination number of G, respectively. For vertices x, y ∈ V(G), we will use deg(x) to denote the degree of x and d(x, y) to denote the distance between x and y. The distance function dG(x) is the sum of distances between x and all other vertices of G. Given a graph G, the branch vertex is a vertex of degree at least 3 and the pendent vertex or leaf is a vertex of degree one. A segment of a tree T is a subpath whose end vertices are branch vertices or leaves, and all internal vertices have degree two. Given a graph G with n vertices, the non-increasing sequence is called the degree sequence of G.
The most well known distance-based index is probably the Wiener index, defined as the sum of distances between all pairs of vertices,Related to the Wiener index is the Wiener Polarity index, WP(T), defined as the number of unordered pairs of vertices of distance 3 in T. Both of them were first introduced in [90] in 1947. Some recent work can be found in [7], [8], [9], [44], [45], [46], [49], [66], [67], [91], [94].
For a tree T and any function f defined on the distance between two vertices of T, a general distance-based graph invariant with respect to f can be defined asWhen this is the classic Wiener index. When this is the Harary index introduced in 1993 and denoted by H(T) [41], [70]. In the same year the hyper-Wiener index of a tree T was proposed in [72] and defined as with
The terminal Wiener index [34] of a tree T was defined as the sum of distances between leaves. This is denoted by where L(T) is the set of pendent vertices of T. A good survey on this index can be found in [33].
Some recent work has also been done on the eccentricity sum of a tree T [78]. That is, where is the eccentricity of a vertex v ∈ V(T).
The Laplacian polynomial P(T, λ) of a tree T is also related to distance-based indices. It is defined as the characteristic polynomial of its Laplacian matrix : It has been shown in [32] that the coefficients and are equal to its Wiener index W(T) and modified hyper-Wiener index WWW(T), respectively. Related work can be found in [24], [25], [39], [40], [64], [82].
The first degree-based index is probably the Randić index [71], defined as One may see summaries of recent results in the survey papers [47], [65]. The generalization of this concept was proposed by Bollobás and Erdős [2] in 1998 and defined as for α ≠ 0, which is also called the general Randić index or the connectivity index [17]. In fact, Randić also introduced which is known today as the Modified Zagreb index. Other examples of generalization of Randić index can be found in [69], [77].
Given a bivariate symmetric function f defined on the degrees of adjacent vertices in T, the connectivity function associated with f [88] is a natural generalization of the above mentioned degree-based indices, defined as
It is easy to see that Rf(T) represents various degree-based indices with different choices of f. For instance, the second Zagreb index is with . Note that the first Zagreb index M1(T) is simply the sum of the squares of degrees [36], [37]. A generalization of M1 is called the Zeroth-order general Randić index, denoted by for α ≠ 0 or 1 [50]. When the function Rf(T) is known as the atom-bond connectivity (or simply the ABC index) ABC(T) [23]. Other examples of such degree-based graph invariants includes the sum-connectivity index [99] with the general sum-connectivity index [100] with the reformulated Zagreb index [97] with and the harmonic index [98] with
Among trees with a given degree sequence the so-called greedy structures have been known to be extremal with respect to many distance-based and degree-based indices. We first list some related definitions. Definition 1.1 (Greedy Trees) Given the degree sequence such that the greedy tree is obtained through the following “greedy algorithm”: Label the vertex with the largest degree as v (the root); Label the neighbors of v as assign the largest degrees available to them such that deg(v1) ≥ deg(v2) ≥ ⋅⋅⋅; Label the neighbors of v1 (except v) as such that they take all the largest degrees available and that deg(v11) ≥ deg(v12) ≥ ⋅⋅⋅, then do the same for ; Repeat (iii) for all the newly labeled vertices, always start with the neighbors of the labeled vertex with largest degree whose neighbors are not labeled yet.
As an example, a greedy tree with degree sequence is shown in Fig. 1.
A caterpillar is a tree whose removal of leaves results in a path. Given a degree sequence the greedy-caterpillar is defined as the following. Definition 1.2 (Greedy Caterpillar) A greedy caterpillar is a caterpillar where the path formed by the internal vertices can be labeled as such thatwhere di is the degree of vertex vi.
In the study of many extremal problems it is convenient to compare the greedy trees of different degree sequences. For this purpose a partial ordering of the degree sequences of trees of fixed order needs to be introduced. Definition 1.3 (Majorization) Given two non-increasing degree sequences π and π′ with and we say that π′ majors π, denoted by π◁π′, if the following conditions are met: for .
Section snippets
Survey of extremal trees with given degree conditions
In this section we survey some of the recent work on extremal trees under various degree conditions with respect to different graph indices. For convenience we let denote a tree with vertices and degree sequence .
In the next a few statements, let and be the set of all trees of order n with exactly b branch vertices and s segments, respectively. Theorem 2.1 Let : ([56]) for the greedy caterpillar
Consequences of Greedy trees and majorization between degree sequences
In this section we first provide some general extremal results among trees with a given degree sequence. Then greedy trees of different degree sequences are compared. Last but not least, these results are applied to various distanced-based and degree-based indices in trees under different degree conditions.
Concluding remark
We provided a brief introduction to distance-based and degree-based graph indices and the associated extremal problems in trees. Recent work on the extremal trees with various degree conditions is surveyed. In particular, greedy trees of a given degree sequence and the comparison between greedy trees of different degree sequences were shown to be very useful tools. Following several key results many extremal trees with degree conditions follow as immediate consequences. These consequences
Acknowledgments
This work is partially supported by the National Natural Science Foundation (61872200), Natural Science Foundation of Tianjin (17JCQNJC00300), the Major Science and Technology Program of Big Data and Cloud Computing of Tianjin (15ZXDSGX00020), the Science and Technology Commission of Tianjin Binhai New Area (BHXQKJXM-PT-ZJSHJ-2017005) and the National Key Research and Development Program of China (2016YFC0400709).
The authors are also in debt to the anonymous referees for their valuable
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