Some Vertex/Edge-Degree-Based Topological Indices of <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2724395pt" id="M1" height="8.14084pt" version="1.1" viewBox="-0.0657574 -7.8684 7.40017 8.14084" width="7.40017pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-115" d="M393 379C402 394 400 411 393 422C384 437 365 448 348 448C301 448 237 372 186 285H182L193 335C210 408 205 448 178 448C150 448 80 402 29 344L45 321C80 355 114 373 122 373C128 373 130 365 124 330C106 228 76 98 50 -5L57 -12C82 -5 112 3 132 6L172 203C196 256 234 304 254 329C275 355 293 367 306 367C318 367 330 360 342 348C347 343 355 343 365 350S386 367 393 379Z"/></g></svg>-</span>Apex Trees

Ali, Akbar; Iqbal, Waqas; Raza, Zahid; Ali, Ekram E.; Liu, Jia-Bao; Ahmad, Farooq; Chaudhry, Qasim Ali

doi:https://doi.org/10.1155/2021/4349074

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Volume 2021 | Article ID 4349074 | https://doi.org/10.1155/2021/4349074

Some Vertex/Edge-Degree-Based Topological Indices of -Apex Trees

Akbar Ali,¹Waqas Iqbal,²Zahid Raza,³Ekram E. Ali,^1,4Jia-Bao Liu ,⁵Farooq Ahmad,^1,6and Qasim Ali Chaudhry^1,7

Academic Editor: Muhammad Kamran Siddiqui

Received11 Aug 2021

Accepted16 Sept 2021

Published21 Oct 2021

Abstract

In chemical graph theory, graph invariants are usually referred to as topological indices. For a graph , its vertex-degree-based topological indices of the form are known as bond incident degree indices, where is the edge set of , denotes degree of an arbitrary vertex of , and is a real-valued-symmetric function. Those indices for which can be rewritten as a function of (that is degree of the edge ) are known as edge-degree-based indices. A connected graph is said to be -apex tree if is the smallest nonnegative integer for which there is a subset of such that and is a tree. In this paper, we address the problem of determining graphs attaining the maximum or minimum value of an arbitrary index from the class of all -apex trees of order , where and are fixed integers satisfying the inequalities and .

1. Introduction

All the graphs discussed in the present paper are finite. The vertex set and edge set of a graph are denoted by and , respectively. Denote by (or simply by if there is no confusion about the graph under consideration) the degree of a vertex . Those graph-theoretical notation and terminology that are used in this paper without defining here can be found in some standard graph-theoretical books, such as [1, 2].

For a graph , its graph invariant is a numerical quantity calculated from by using any rule in such a way that the equation holds for every graph isomorphic to . In chemical graph theory, graph invariants are usually referred to as topological indices [3–10]. A topological index of a graph that depends on the degrees of the vertices of is known as a vertex-degree-based topological index; similarly, edge-degree-based topological indices are defined. To the best of the present authors’ knowledge, the Platt index [11, 12] is the oldest vertex-degree-based topological index; for a graph , its Platt index is defined as

Since is degree of the edge , the Platt index is also an edge-degree-based topological index.

In the present paper, we are concerned with the following type of vertex-degree-based topological indices:which are known as bond incident degree () indices (see, for example, [13]), where is a real-valued-symmetric function. Those indices for which can be rewritten as a function of are known as edge-degree-based indices. Note that the Platt index defined in equation (1) is a vertex/edge-degree-based index. Other examples of indices include the first Zagreb index [14], second Zagreb index [15], general Randić index [16, 17], general zeroth-order Randić index [17, 18], general sum-connectivity index [19], natural logarithm of the multiplicative second Zagreb index [20], variable sum exdeg index [21], sum lordeg index [21], augmented Zagreb index [22], general Platt index [23], and Sombor index [24]. The choices of the function that correspond to the aforementioned indices are specified in Table 1.

In order to solve an extremal problem concerning the topological index (which is same as the second Zagreb index, see Table 1), Bollobás et al. [25] considered following generalization of the general Randić index of a graph :by taking as any real number and as any nonnegative integer. We note that the graph invariant (3)(i)Remains well-defined if is any real number greater than (ii)Gives the reduced second Zagreb index [11] when one takes and (iii)Coincides with the variable connectivity index [26–28] if and is any nonnegative real number

Thus, in what follows, we assume that and call the graph invariant (3) as the Bollobás—Erdős—Sarkar index and denote it by , where is the set of all real numbers greater than , is the set of all real numbers, is the set of all positive real numbers, and . Thus, the Bollobás—Erdős—Sarkar index of a graph is defined aswith . Certainly, the Bollobás —Erdős—Sarkar index is a index (here, it needs to be mentioned that the graph invariant was defined in [29] for any real number ).

A connected graph is said to be -apex tree if is the smallest nonnegative integer for which there is a subset of such that and is a tree. (Unfortunately, the terminology of apex trees and -apex trees, being used by many researchers particularly in chemical graph theory, may arise confusion with the terminology of apex graphs and -apex graphs, respectively. According to Mohar [30], a graph is an apex graph if it contains a vertex such that is planar. Also, according to Thilikos and Bodlaender [31], a graph is an -apex graph if it can be made planar by removing at most vertices.) The set is known as -apex set and its members are known as apex vertices. Every tree is a 0-apex tree. (Throughout this paper, whenever we consider a class of graphs of the same order, we assume that all the graphs of the considered class are pairwise nonisomorphic.) In this paper, we address the problem of determining graphs attaining the maximum or minimum value of an arbitrary index from the class of all -apex trees of order , where and are fixed integers satisfying the inequalities and .

2. Main Results

The join of two graphs and is the graph with the vertex set and the edge set . If is not an edge in a graph , then denotes the graph formed by adding the edge in . The complete graph and the star graph of order are denoted by and , respectively. To state and prove the first main result, we need the following known result.

Lemma 1 (see [32]). Let be a topological index.(i)If for every connected noncomplete graph , the inequality holds for every ; then, the graph attaining the maximum value of the topological index among all -apex trees of order is isomorphic to the join , where and are fixed integers satisfying the inequalities and and is a tree of order .(ii)If for every connected noncomplete graph , the inequality holds for every ; then, the graph attaining the minimum value of the topological index among all -apex trees of order is isomorphic to the join , where and are fixed integers satisfying the inequalities and and is a tree of order .

For a graph and a vertex , denote by the set of all those vertices of that are adjacent to . Now, we state and prove our first main result.

Theorem 1. Let be the set of all real numbers. Let be a real-valued-symmetric function such that(i)The inequality holds for and (ii)Both and are increasing in , where denotes the partial derivative of with respect to (iii)The function satisfies at least one the following additional conditions: is strictly increasing; If is a bond incident degree index such that, for every connected noncomplete graph , the inequality holds for every ; then, uniquely attains the maximum index among all -apex trees of order , where and are fixed integers satisfying the inequalities and .

Proof. Let be a graph attaining the maximum index in the given class of graphs. From Lemma 1, it follows that is the join of the complete graph and a tree of order . It remains to prove that . Suppose to the contrary that . Let be a vertex of maximum degree in T. Then, there exist vertices such that is a path in . Take . Let be the graph deduced from by deleting the edges and adding the edges . Observe that the graph remains an -apex tree of order . In the remaining proof, by the vertex degree , we mean degree of the vertex in the graph . Now, by using the definition of the index and the constraints on the function , we getSince is increasing, the right hand side of (5) is nonnegative, which contradicts our assumption that attains the maximum index among all -apex trees of order .
Since every function satisfies all the conditions of Theorem 1, with and , the next result is an immediate consequence of Theorem 1.

Corollary 1. Among all -apex trees of order , the join uniquely attains the maximum values of the Sombor index, general sum-connectivity index , general Platt index , and variable sum exdeg index , where , , and and are fixed integers satisfying the inequalities and .
The extremal result concerning the general sum-connectivity index mentioned in Corollary 1 was proven by using some other way: in [33], for and ; in [34, 35], for and ; in [36], for and . Also, the result concerning mentioned in Corollary 1 was proven in [37] for by other means. Moreover, the result concerning the topological index mentioned in Corollary 2 was proven by using some other way in [38] for .
Since the proof of the next result is fully analogous to that of Theorem 1, we omit it.

Theorem 2. Let be the set of all real numbers. Let be a real-valued-symmetric function such that(i)The inequality holds for and (ii)Both and are decreasing in , where denotes the partial derivative of with respect to (iii)The function satisfies at least one the following additional conditions: is strictly decreasing; If is a bond incident degree index such that, for every connected noncomplete graph , the inequality holds for every ; then, uniquely attains the minimum index among all -apex trees of order , where and are fixed integers satisfying the inequalities and .
Theorems 1 and 2 can be improved if one considers the indices of the following form:where , is a strictly increasing function, and is a strictly decreasing function (where denotes the derivative of ).

Theorem 3. Let be the set of all real numbers. For , let be a real-valued symmetric function. Also, let be strictly increasing and be strictly decreasing, where denotes the derivative of . Let such that, for every connected noncomplete graph , the inequality(i) holds for every ; then, uniquely attains the maximum value of the index among all -apex trees of order , where and are fixed integers satisfying the inequalities and (ii) holds for every ; then, uniquely attains the minimum value of the index among all -apex trees of order , where and are fixed integers satisfying the inequalities and

Proof. We prove part (i) of the theorem. Part (ii) can be proved in a fully analogous way. Let be a graph attaining the maximum value of the index in the given class of -apex trees. From Lemma 1, it follows that is the join of the complete graph and a tree of order . It remains to prove that . Suppose to the contrary that . Let be a vertex of maximum degree in T. Then, there exist vertices such that is a path in . Take . Let be the graph deduced from by deleting the edges and adding the edges . Observe that the graph remains an -apex tree. In the remaining proof, by the vertex degree , we mean degree of the vertex in the graph . Here, we haveBy Lagrange’s mean value theorem, there exist real numbers and such thatandThe inequality gives , which implies that the right hand side of equation (9) is negative, because is strictly increasing. Thus, we have , which contradicts our assumption that attains the maximum value of the index among all -apex trees of order .
The next result follows directly from the first part of Theorem 3.

Corollary 2. Among all -apex trees of order , the join uniquely attains the maximum values of the general zeroth-order Randić index for , multiplicative second Zagreb index , and sum lordeg index, where and are fixed integers satisfying the inequalities and .

Proof. It is clear that any graph has the maximum value in a given graph class if and only if has the maximum value in the considered graph class. Define the functions with and , with , and with (see [39]). Observe that, for every , the derivative function of is strictly increasing. Hence, the desired result now follows from Theorem 3.

Remark 1. The result concerning the general zeroth-order Randić index mentioned in Corollary 3 was proven by using some other way: in [40] for and ; in [41] for and ; in [42] for and .
For proving our next result, we need the following known result.

Lemma 2 (see [32]). Let be a topological index.(i)If for every connected noncomplete graph, the inequality holds for every ; then, the graph attaining the minimum value of the topological index among all 1-apex trees of a fixed order is a unicyclic graph, and its unique cycle has a vertex of degree 2.(ii)If for every connected noncomplete graph, the inequality holds for every ; then, the graph attaining the maximum value of the topological index among all 1-apex trees of a fixed order is a unicyclic graph, and its unique cycle has a vertex of degree 2.

Note that, for the general zeroth-order Randić index , it holds that, for every connected noncomplete graph , one hasfor every . Also, note that the class of all (connected) unicyclic graphs forms a subclass of the class of all 1-apex trees. Moreover, in [43], it was proven that among all unicyclic graphs of a fixed order , the graph formed by adding an edge in the star attains the maximum general zeroth-order Randić index for , attains the minimum general zeroth-order Randić index for , and the cycle graph attains the minimum general zeroth-order Randić index for . Thus, keeping in mind these observations and Lemma 5, one gets the next result.

Corollary 3. Among all 1-apex trees of a fixed order , the graph formed by adding an edge in the star attains the maximum general zeroth-order Randić index for , attains the minimum general zeroth-order Randić index for , and the cycle graph attains the minimum general zeroth-order Randić index for .

We remark here that Corollary 3 We remark here that Corollary 6was proven in [40] by using some other way.

Next, we derive a result about the augmented Zagreb index of 1-apex trees. For this, we need the following lemma first.

Lemma 3 (see [44]). For every fixed integer , the graph formed by adding an edge in the star uniquely attains the minimum in the class of all unicyclic graphs with vertices, and the minimum value is

Since for every connected noncomplete graph , it holds that for every (see [44]), and the next result follows from Lemmas 5 and 7.

Theorem 4. For every fixed integer , the graph formed by adding an edge in the star uniquely attains the minimum in the class of all 1-apex trees of order , and the minimum value isTheorem 4 was proven in [45] by using some other way.
Finally, we determine the unique graph attaining the maximum value of . For this, we need the following two results concerning the Zagreb indices of -apex trees.

Lemma 4 (see [34, 35]). If is an -apex tree of order , then it holds thatwith equality if and only if , where and .

Lemma 5 (see [34, 35]). If is an -apex tree of order , then it holds that with equality if and only if , where and .

From the following identity,

Lemmas 4 and 5, the next result follows.

Theorem 5. In the class of all -apex trees of order , the join uniquely attains the maximum -value, where and are fixed integers satisfying the inequalities and and is any nonnegative real number. In other words, if is an -apex tree of order , then it holds thatwith equality if and only if .

Theorem 5 remains true if one replace the condition “ is any nonnegative real number” with “ is any real number greater than or equal to .” To prove this modified statement of Theorem 5, we cannot use identity (15) because of the negative values of . In what follows, we prove the aforementioned statement (Theorem 6) by using some other way. For this, we need some additional lemmas first.

Lemma 6. Let and be two nonadjacent vertices of a graph . The inequality holds for every real number greater than . Also, it holds that

Proof. The result immediately follows from the definition of .

Lemma 7 (see [46, 47]). If is a tree of order , then it holds thatwith equality if and only if .

Lemma 8 (see [48]). If is a tree of order , then it holds thatwith equality if and only if .

Now, we are able to state and prove our final result.

Theorem 6. In the class of all -apex trees of order , the join uniquely attains the maximum -value, where and are fixed integers satisfying the inequalities and and is any real number greater than or equal to . In other words, if is an -apex tree of order , then it holds thatwith equality if and only if .

Proof. Suppose that is a graph attaining the maximum -value in the given class of graphs. From Lemmas 1 and 6, it follows that the graph is isomorphic to the join , where is a tree of order . Let be a vertex of degree . Note that the size of the graph isThus, the size of isAlso, one hasWe note that the vertex is an apex vertex and the graph is an -apex tree of order . If , then one gets the desired result by using Lemmas 7 and 8 in equation (23). If , then one gets the desired result by using Lemmas 4 and 5 in equation (23).
The next result about the reduced second Zagreb index is a special but notable case of Theorem 6.

Corollary 4. If is an -apex tree of order , then it holds thatwith equality if and only if , where and .

Data Availability

The data used to support the findings of the study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was funded by Scientific Research Deanship at University of Ha’il, Saudi Arabia, through project no. RG-20 050.

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