Some Inequalities between General Randić-Type Graph Invariants

Abstract

The Randić-type graph invariants are extensively investigated vertex-degree-based topological indices and have gained much prominence in recent years. The general Randić and zeroth-order general Randić indices are Randić-type graph invariants and are defined for a graph with vertex set as and , respectively, where is an arbitrary real number, denotes the degree of a vertex , and represents the adjacency of vertices and in . Establishing relationships between two topological indices holds significant importance for researchers. Some implicit inequality relationships between and have been derived so far. In this paper, we establish explicit inequality relationships between and . Also, we determine linear inequality relationships between these graph invariants. Moreover, we obtain some new inequalities for various vertex-degree-based topological indices by the appropriate choice of .

1. Introduction

In this paper, we consider a simple finite graph with the vertex set and the edge set , where the quantities and are known as the order and the size of , respectively. If , then is called a nontrivial graph. The ceiling function would round to the smallest integer greater than or equal to , whereas the floor function would round to the largest integer less than or equal to . For a given vertex , the neighborhood of is denoted by and defined as , where represents the adjacency of vertices and in . For , the degree of the vertex is defined as . Among all vertices of , the maximum degree is given by and the minimum degree is given by . Without loss of generality, the degree sequence of the vertices in is organized as . If for each vertex in , we call it a regular graph. For a vertex , denote by . It is obvious that and .

A chemical (or molecular) graph can frequently be used to represent the structure of a molecule. Chemical graphs play a pivotal role in understanding and representing the structural intricacies of molecules, thereby serving as a fundamental tool in the realm of chemistry. These graphs, composed of vertices representing atoms and edges denoting chemical bonds, provide a visual abstraction that aids in deciphering the three-dimensional arrangement of atoms in a compound. The chemical importance of graphs lies in their ability to elucidate molecular properties, reactivity patterns, and overall structural characteristics critical for predicting a substance’s behavior. A plethora of literature exists, delving into the development and application of various graph-based approaches in chemistry [1]. Graph theory has proven invaluable in medicinal chemistry, material science, and computational chemistry, offering insights into molecular relationships, reaction mechanisms, and the rational design of novel compounds [2].

Graph theory has contributed to the development of chemistry by providing a variety of mathematical tools such as topological indices. A graph invariant that is calculated from the parameters of a chemical graph is declared a topological index (TI) if it correlates with some molecular property. TIs are the conclusive results of a mathematical and logical procedure that maps the chemical phenomena hidden inside a molecule’s symbolic representation into a useful value, and they have been shown to be useful in modelling varied physicochemical characteristics reflected by QSAR and QSPR calculations [3, 4].

Milan Randić, a chemist, proposed a degree-based topological index, called the Randić index [5] which is useful for measuring the degree of branching in the carbon-atom skeleton of saturated hydrocarbons. This index is represented by and is defined as follows:

Randić proved that this index is significantly associated with a variety of physicochemical features of alkanes, including boiling points, enthalpy of formation, surface areas, and chromatographic retention times [6, 7]. Eventually became one of the most well-known molecular descriptors, with two books [8, 9], several reviews, and a plethora of research articles devoted to it. Some bounds of this index have been studied in [10]. Bollobás and Erdös [11] extended by substituting an arbitrary real number for the exponent . This graph invariant is called the general product-connectivity index or the general Randić index [12], represented by :

Kier and Hall [13] put forward the zeroth-order Randić index, represented by . The explicit formula of is

Eventually, Li and Zheng [14] proposed zeroth-order general Randić index by replacing the fraction by an arbitrary real number different from 0 and 1, denoted by :

This index is also studied under the name first general Zagreb index [15]. Moreover, it may be noted that and are also studied under the names first Zagreb index [16] and second Zagreb index [17], respectively. The AutoGraphiX (conjecture-generating computer method) proposed [18] that the Zagreb indices are generally related to the inequality for a connected graph with order and size . Though there exist graphs for which it does not hold [19], it is true for numerous classes of graphs [20–23].

The investigation of relationships between two topological indices remains an intriguing and attractive problem for researchers. Liu and Gutman [24] derived the implicit inequalities between and for and . Later, Zhou and Vukičević [25] established the inequalities between , , , and . In this paper, we make a step forward by deriving the explicit relationships between and for and . Also, we obtain linear inequalities between and for and , for alpha and with some condition on the order of graph for alpha . Moreover, we obtain new inequality between and for any graph with order and size . Further, we determine new inequality between and .

2. Some Known Results

In this section, we review some known results that will be used in our main results.

Let and be positive real numbers such that for , it holds that and . Then,where . Further, equality attains if and only if and [26].

Rodríguez et al. [27] established the following relationships between and .

Let be a nontrivial graph having the parameters and . Then, for ,and for ,

Equality attains in each case if and only if is regular.

Also, Rodríguez et al. [27] derived the following relation between and .

If is a nontrivial graph with the parameters , , and , then for ,

Further, equality attains if and only if is regular.

Liu and Gutman [24] derived the following implicit quadratic inequality between and .

If is a nontrivial graph with the parameters , , and , then for ,and the equality is achieved if and only if is regular.

Also, Liu and Gutman [24] established the following inequality between , , , and .

If is a nontrivial graph having the parameters and , then for ,

Further, equality attains if and only if is regular.

3. Main Results

Lemma 1. Let be a nontrivial graph; then, for any real number ,where .

Proof. By rearranging with respect to the sum of degrees of neighbor vertices of each vertex , we haveBy setting , the required result follows.

Lemma 2. Let be a nontrivial graph; then, for any real number ,where.

Proof. By taking , the desired result follows.
In the following theorem, we derive the left and right explicit inequalities between and for and , respectively.

Theorem 3. Let be a nontrivial graph with order , size , minimum vertex-degree , and maximum vertex-degree . Then, the following left and right inequalities hold for and , respectively:where and . Further, each equality holds if and only if is a regular graph.

Proof. We choose and in inequality (1); then, for , and and for , and , where . Note that for any real number , . Then, for any real number , inequality (1) takes the formwhere .
From (11) and (14), we haveThis implies thatThis givesBy using (6) with and (7) with in the left and right inequalities, respectively, we haveBy taking , the required inequality (16) follows.
Since equality attains in (5) if and only if and , this gives that equality attains in (16) if and only if and . Also, and imply that and . This recommends that each equality in (16) attains if and only if is a regular graph.
In the following corollary, we derive the linear inequality between and for any positive real number .

Corollary 4. Let be a nontrivial graph having order , size , minimum vertex-degree , and maximum vertex-degree with . Then, for , we havewhere and . Moreover, equality attains if and only if is a regular graph.

Proof. From inequality (9) with , we haveAlso, from left inequality (16), we getwhere and .
By using inequality (24) in inequality (23), we haveAfter simplifying (25) and then rearranging, we get the desired inequality (22).
Since the equality attains in both inequality (9) and left inequality (16) if and only if is a regular graph, equality in (22) holds if and only if is a regular graph.

Lemma 5. For , it is easy to observe thatwhere equality attains if and only if is a regular graph.

In the upcoming corollary, we derive the linear inequality between and for any negative real number which satisfies some condition on the order of graph .

Corollary 6. Let be a nontrivial graph having order , size , minimum vertex-degree , and maximum vertex-degree . Then, for and ,where ,and. Further, equality attains if and only if is a regular graph.

Proof. From inequalities (4) and (14) with , inequality (6) becomesBy taking and rearranging, we haveAlso, from right inequality (9) with and , we getwhere and .
After simplifying (30), we achieve the desired inequality (27).
Since equality attains in right inequality (16) and each of the inequalities (4), (14), and (6) if and only if is a regular graph, this implies that equality in (27) attains if and only if is a regular graph.
In the following corollary, we get a new inequality between and for any graph with order and size by taking in the left inequality (16).

Corollary 7. Let be a nontrivial graph with order and size ; then,where and . Further, each equality holds if and only if is a regular graph.

In the following corollary, we obtain a new inequality between and for any graph with order and size by setting in the right inequality (16).

Corollary 8. Let be a nontrivial graph having order and size ; then,where and . Moreover, each equality holds if and only if is a regular graph.

4. Conclusions

A major contribution of this paper lies in the derivation of explicit inequality relationships between the general Randić and zeroth-order general Randić indices. Furthermore, the paper goes beyond the implicit relationships and determines linear inequality relationships between the general Randić and zeroth-order general Randić indices, providing a more comprehensive framework for their comparison. We would like to conclude this paper by proposing the following open problem.

Open Problem 9. Drive the linear inequality between the general Randić index and zeroth-order general Randić index for any negative real number .

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.

Acknowledgments

Open-access funding was enabled and organized by JISC.