The Third Leap Zagreb Index for Trees

Abstract

The third leap Zagreb index is the sum of the products of vertex degrees and second degrees. In this paper, a lower bound on the third leap Zagreb index is established, and the extremal trees achieving this bound are characterized.

1. Introduction

In this paper, we consider simple and connected graphs. Let G be such a graph with vertex set and edge set . The order of G is denoted by and the size of G is denoted by . The degree of a vertex in G is the number of edges incident to and is denoted by . The minimum and maximum degrees of a graph G are denoted by and , respectively.

The distance between any two vertices u and of a graph G is equal to the length of a shortest path connecting them. For a vertex and a positive integer k, the open k-neighborhood of in the graph G, denoted by , is defined as . The k-distance degree of a vertex in G, denoted by , is the number of k-neighbors of the vertex in G, i.e., . Evidently, for every .

A leaf of a tree T is a vertex of degree one. A stem is a vertex adjacent to a leaf, and a strong stem is a stem adjacent to at least two leaves. An end stem is a stem whose all neighbors with exception at most one are leaves. Denote by the number of leaves that are adjacent to the vertex . A rooted tree is a directed tree having a distinguished vertex ω, called the root. For a vertex in a rooted tree T, let denote the set of children of and denote the set of descendants of , whereas the depth of , , is the largest distance from to a vertex in .

The Zagreb indices are the oldest vertex-degree-based graph invariants. They were introduced in the 1970s [1, 2]. Details of their mathematical theory and chemical applications can be found in the surveys [3–12].

For a graph G, the first and second Zagreb indices are defined asNote that .

In recent years, some novel variants of Zagreb indices have been put forward, such as Zagreb coindices [13, 14], reformulated Zagreb indices [15, 16, 17], Zagreb hyper indices [18, 19], multiplicative Zagreb indices [20], general Zagreb indices [21], multiplicative sum Zagreb indices [22, 23], multiplicative Zagreb coindices [24], etc.

Naji et al. [25] extended the concept of the Zagreb index to the second vertex degrees, conceiving the so-called leap Zagreb indices. For a graph G, the first, second, and third leap Zagreb indices are defined as follows:

In [25], the basic properties of these invariants were established, including bounds in terms of Zagreb indices, order, and the size of the underlying graph. In this paper, we establish a lower bound on the third leap Zagreb index and characterize the extremal trees achieving this bound.

2. A Lower Bound on the Third Leap Zagreb Index of Trees

In this section, we present a sharp lower bound for the third leap Zagreb index of trees in terms of their order and maximum degree and characterize all extremal trees.

Throughout this section, T denotes a rooted tree with root ω, where ω is a vertex of maximum degree and .

We start with some lemmas.

Lemma 1. Let T be a tree of order n with maximum degree . If T has a vertex u of degree at least three in maximum distance from ω, then there is a tree of order n with maximum degree such that .

Proof. Let be a stem of T with and let , where is the parent of u. Assume . We consider the following cases. Case 1. All neighbors of u except are leaves. Subcase 1. is adjacent to a leaf z, and the variable is preceded by a variable. Let be the tree obtained from by attaching the path . Clearly, is a tree of order n with and , , , , , and for . By definitions, we have Subcase 2. has no new leaf neighbor. Let . Then, If , then . If and , then . If and , then . Case 2. u is adjacent to a leaf , and is a path in T for . If is a stem, then the result is immediate as in Case 1. So assume that is not a stem. Let be the tree obtained from by attaching the path . Then, , , , , and for . Let . Then, Case 3. u is not a stem, and and are two paths in T. Clearly , , and for . Let be the tree obtained from by attaching the path . Then, , , and for . Also, we have for , , , for , , and . Now, we consider the following subcases. Subcase 1. . Subcase 2. . Subcase 3. .A spider is a tree with at most one vertex of degree greater than 2, called the center of the spider. If there is no vertex of degree greater than two, then any vertex can be the considered as the center. A leg of the spider is a path from the center to a vertex of degree one. Thus, a star with k edges is a spider with k legs, each of length 1, and the path is a spider with 1 or 2 legs.
By Lemma 1, among all trees of order n with maximum degree , the spiders have the minimum third leap Zagreb index. In what follows, we determine the spiders having minimum third leap Zagreb index. If , then . Therefore, let .

Lemma 2. Let T be a spider of order n with legs. If T has at least two legs of length at least 2, then there is a spider of order n with p legs such that .

Proof. Let ω be the center of T and let , be two legs of length at least two in T. Let be the tree obtained from by attaching the path , and let . We consider the following cases. Case 1. . Case 2. . Case 3. .Now, we are ready to state our main result.

Theorem 3. For any tree T of order with maximum degree ,with equality if and only if T is a spider with at most one leg of length at least two.

Proof. Let be a tree of order with maximum degree such thatLet be a vertex with maximum degree . Root at . By the choice of , we deduce from Lemma 1 that is a spider with center . It follows from Lemma 2 and the choice of that has at most one leg of length at least two. First, let all legs of have length one. Then, is a star of order n and . Now, let have only one leg of length at least two. Then, when , and when . This completes the proof.

3. Conclusion

We continue the study of the third leap Zagreb index and establish the lower bound on the third leap Zagreb index of trees in terms of their order and maximum degree. Also, we characterize the extremal trees achieving this bound. We conclude this paper with two open problems.

Problem 1. Present sharp lower bounds for the first and second leap Zagreb indices of trees in terms of their order and maximum degree.

Problem 2. Present sharp upper bounds for the leap Zagreb indices of trees in terms of their order and maximum degree.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This study was supported by Humanities and Social Sciences Research Project of the Ministry of Education “Research on the Development of Chinese Family Policies at Low Fertility Levels” (No. 19YJCZH069) and Anhui Education Department Teaching and Research Fund Project (No. 2018jyxm1305); Anhui University of Finance and Economics School-Level Teaching and Research Fund Project (acxkjsjy201803zd and acjyyb2018006); and Chizhou University Teaching Team Project (2016XJXTD02).