On the extremal values for the Mostar index of trees with given degree sequence
Introduction
In this paper, all the graphs we considered are simple and undirected. Let be a graph with vertex set VG and edge set EG. Unless otherwise stated, we follow the traditional notations and terminologies (see [3]).
For a set X, denote by |X| its cardinality. Thus, |VG| is called the order of G. For v ∈ VG, denote by dG(v) (or d(v) for short) the degree of v. Denote by N(v) the set of vertices adjacent to v. And denote by E(v) the set of edges incident with v. The distance between u, v in G is the least length among all paths in G, which is denoted by dG(u, v) (or d(u, v) for short). The distance between and w in G is defined as min {d(u, w), d(v, w)}, which is denoted by dG(e, w) or dG(w, e) (or d(e, w), d(w, e) for short). For each edge letAnd let (or put for short), for . Note that . And for each e ∈ EG if and only if G is bipartite. Specially, a graph G is called distance-balanced if for each edge uv ∈ EG. One may be referred to [2], [6], [13], [14], [15] and the references cited therein, for the study on distance-balanced graph invariants. Since there exist many graphs which are not distance-balanced, to measure how far is a graph from being distance-balanced is a natural problem. However, such measuring invariant was proposed only recently, in 2018, by Došlić et al. [8], and was named by Mostar index, which is defined aswhereis called the contribution of the edge uv for Mo(G).
Clearly, a graph G is distance-balanced if and only if . The Mostar index produces a global measure of peripherality of G by calculating the sum of peripherality contributions over all edges in G. In [8], Došlić et al. determined the extremal values of the Mostar indices among trees and unicyclic graphs, respectively. And they stated some extremal problems on Mostar index. After that, Tepeh [18] characterized the bicyclic graphs with extremal Mostar index. Hayat and Zhou [10] gave a sharp upper bound of the Mostar index for cacti of order n with k cycles, and characterized all the cacti that achieve this bound. Hayat and Zhou [11] also studied the Mostar index of trees with parameters. For example, they identified those trees with the least Mostar index with fixed order and fixed maximum degree, and those trees with the greatest Mostar index with fixed order and with fixed diameter. Huang, Li and Zhang [12] established sharp upper and lower bounds on the Mostar indices among hexagonal chains with a given number of hexagons are determined, respectively. All the corresponding extremal hexagonal chains are characterized. The authors of the current paper [5] determined the tree-type hexagonal systems (catacondensed hydrocarbons) with the least and the second least Mostar indices are determined.
Very recently, the authors of the current paper [4] determined those chemical trees (trees with the maximum degree at most 4) of order n with the greatest Mostar index, those chemical trees of order n and diameter d with the greatest Mostar index, and those general trees of order n and diameter d with the least Mostar index. Note that [4, Theorems 1.3 and 1.4] can be easily generalized to those trees with maximum degree at most Δ for Δ ≥ 3, by a similar discussion.
The degree sequence of a tree is the sequence of the degrees (in descending order) of the non-leaf vertices. The length of a degree sequence is the number of integers it contains. This paper focuses on the following natural extremal problem. Problem 1 Find extremal trees with a given degree sequence with respect to the Mostar index.
Let be a positive integer sequence where k1 ≥ k2 ≥ ⋅⋅⋅kp ≥ 2 and p ≥ 2. Let be the set of trees which admit K as a degree sequence. Note that all trees in have the same number of verticesFor a graph set let (resp. ) be the set of graphs among having the greatest (resp. least) Mostar index. And for short, let (resp. ) be the Mostar index of each graph in (resp. ).
The first aim of this paper is to determine and . For this aim, we construct a series of trees as follows:
- (1)
(that is a star with exactly k1 leaves) with v0 being the unique non-leaf vertex.
- (2)
For 2 ≤ i ≤ p, is obtained from by attaching pendant edges to some leaf in which has the least distance from v0.
The obtained above is called a nice tree of K which is also denoted by TKnice; see Fig. 1 for example. By definition, is a nice tree of Ki () where . The distance from a vertex v (resp. edge e) to v0 in TKnice is called the height of v (resp. e) in TKnice, and the greatest distance from a leaf to v0 in TKnice is called the height of TKnice which is denoted by h(TKnice) (or h for short). Denote by Lj(TKnice) or Lj for short (resp. ELj(TKnice) or ELj for short) the set of vertices (resp. edges) of height j (0 ≤ j ≤ h), which is called the j-th level (resp. edge level) of TKnice. Let (0 ≤ j ≤ h). Note that TKnice may be not unique. Let be the set of nice trees of K. However, all trees in have the same height h(K) and the same number lj(K) (0 ≤ j ≤ h) of vertices in the j-th level (what is more, the same degree sequence within the j-th level), by the definition. Note that and for 2 ≤ j ≤ h, where (). Letfor the degree sequence K. Theorem 1.1 Let T be in with and p ≥ 2. Assume n(K) and ρ(K) are defined in (1) and (2), respectively. Thenwith equality if and only if .
Let Δ ≥ 2 (resp. or 1 ≤ z ≤ ⌊n/2⌋) be the set of trees of order n with the maximum degree Δ (resp. x leaves, independence number y, or matching number z), where the independence number (resp. matching number) of a tree is the cardinality of the maximum independent set (resp. maximum matching). From Theorem 1.1, we derive the following corollaries which contain some of those results obtained in [11]. Corollary 1.2 Let K1, K2 be two degree sequences satisfying . If ρ(K1) ≥ ρ(K2), then Mo(T1) ≤ Mo(T2) where (), with the equality only if . (where ) consists of nice trees of order n with degree sequences ’s satisfying for 2 ≤ kj ≤ Δ for and . Hayat and Zhou [11] consists of nice trees of order n with degree sequence . consists of nice trees of order n with degree sequence . Hayat and Zhou [11] consists of nice trees of order n with degree sequence .
The second aim of this paper is to give some properties of trees in though is not completely determined yet. A caterpillar is a tree whose non-leaf vertices exactly induces a path (the path is called the spine of the caterpillar). Let be the set of caterpillars in . Let such that (j ∈ [1, p]) where is the spine of . A tree T of order n is called v*-central (or central for short), if there exits a vertex v* such that each component of contains less than n/2 vertices and v* is called the center of T; see Fig. 2(a) for example. A tree T of order n is called e*-edge central (or edge central for short) if there exits an edge (e* is called the edge center and are called the twin centers) such that each of the two components of contains exactly n/2 vertices (so n is even); see Fig. 2(b) for example. Note that each tree is either central or edge central [9]. Property 1.3 Let T be in with a being the center or edge center, where (k1 ≥ k2 ≥ ⋅⋅⋅ ≥ kp ≥ 2, p ≥ 2). If u, v are two non-leaf vertices in the same extended branch of T with d(u, a) > d(v, a) (v is not the center when T is central), then dT(u) ≥ dT(v). T is isomorphic to a caterpillar . What is more, one has and for some . If T is v*-central and v* is not a spine end, then at most one of the two spine neighbors of v* has the degree less than v*.
Note that, for each tree one hasSo Property 1.3(2) implies that to determine and it is sufficient to determine the permutation of such that and for some to make (3) minimal. However, such permutation respect to the minimal value of (3) seems difficult to determine in general, since there are non-trivial bi-partitions of in the worst case. From the point of the structures of trees, we determine and when . Theorem 1.4 Let T be in with a being the center or edge center, where (k1 ≥ k2 ≥ ⋅⋅⋅ ≥ kp ≥ 2, p ≥ 2). If thenwith equality if and only if . If thenwith equality if and only if .
The last aim of this paper is to study some numerical results. Further linear regression analysis of the distance based indices with some chemical properties of octane isomers and boiling points of benzenoid hydrocarbons is carried out. The linear model, based on the Mostar index, is better than or as good as the models corresponding to the other distance based indices.
In Section 2, we give some preliminaries and the properties of some graph transformations. Theorem 1.1 and Corollary 1.2 are proved in Sections 3. Property 1.3 and Theorems 1.4 are proved in Sections 4. The proofs are based on the properties of moving operations which are stated in Section 2, and which generalize those obtained in [4]. In Section 5, applications of Mostar index to octane isomers and benzenoid hydrocarbons are given. Summary and conclusion are given in the last section.
Section snippets
Preliminaries and the properties of some graph transformations
In this section, some necessary definitions and preliminary properties are given.
Let T be a tree. For u, v ∈ VT, recall that there is a unique path (denoted by Pu, v) in T. If T is v*-central, let and for vi ∈ N(v*). Then has exactly d(v*) connected components. The connected component containing vi is called the vi-branch of T, which is denoted by for vi ∈ N(v*). And is called the vi-extended branch of T for i ∈ [1, d(v*)]. If T is e
Greatest Mostar index in
In this section, we prove our first main result, which determines and . First we give some useful lemmas. Lemma 3.1 Let T be in where (k1 ≥ k2 ≥ ⋅⋅⋅ ≥ kp ≥ 2, p ≥ 2). If T is v*-central, then d(u) ≥ d(v) for each u ∈ Rj and (j ≥ 0). Proof Let u ∈ Rj and (j ≥ 0). Suppose d(u) < d(v). Then we can do moving operation on (F, u) where F⊆[E(v)∩ES(v)] and to increase the Mostar index by Property 2.1(i) and (ii), with the newly obtained tree being also in
Least Mostar index in
In this section, we prove some properties of trees in . Proof of Property 1.3 For (1), let u, v be two arbitrary vertices which are in the same extended branch of T with d(u, a) > d(v, a) (v is not the center when T is central). Suppose dT(u) < dT(v). Then we can do moving operation on (F, u) where and to decrease the Mostar index by Property 2.1(i), a contradiction. So dT(u) ≥ dT(v). So (1) holds. For the first part of (2), the conclusion of (1) implies that each extended branch
Applications of Mostar index to octane isomers and benzenoid hydrocarbons
In this section we investigate the correlations of some chemical properties of octane isomers and boiling points of benzenoid hydrocarbons with the Mostar index. We also report a comparative study of the Mostar index with other distance-based topological indices, such as Wiener index [20], the first eccentric connectivity index and second eccentric connectivity index [1], [19] and the first status connectivity index and second status connectivity index [17]. All the experimental values of the
Summary and conclusion
In this paper, we consider the problem: How to determine extremal trees with a given degree sequence with respect to the Mostar index. We establish sharp upper bound on the Mostar index of trees with given degree sequence. All the corresponding extremal graphs (called nice trees) are identified. Based on this main result, it is natural and interesting to determine the trees with maximum degree (resp. given number of leaves, given independence number, given matching number) having the largest
CRediT authorship contribution statement
Kecai Deng: Methodology, Software, Writing - original draft. Shuchao Li: Conceptualization, Supervision, Investigation, Validation, Writing - review & editing.
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