Mathematical Models and Computation in Discrete DynamicsView this Special Issue
Minimum Variable Connectivity Index of Trees of a Fixed Order
The connectivity index, introduced by the chemist Milan Randić in 1975, is one of the topological indices with many applications. In the first quarter of 1990s, Randić proposed the variable connectivity index by extending the definition of the connectivity index. The variable connectivity index for graph is defined as , where is a nonnegative real number, is the edge set of , and denotes the degree of an arbitrary vertex in . Soon after the innovation of the variable connectivity index, its various chemical applications have been reported in different papers. However, to the best of the authors’ knowledge, mathematical properties of the variable connectivity index, for , have not yet been discussed explicitly in any paper. The main purpose of the present paper is to fill this gap by studying this topological index in a mathematical point of view. More precisely, in this paper, we prove that the star graph has the minimum variable connectivity index among all trees of a fixed order , where .
All the graphs that we discuss in the present study are simple, connected, undirected, and finite. For a graph , the number is called its order, and is the size of . Neighbor of a vertex is a vertex adjacent to . The set of all neighbors of vertex of is denoted by . The number is called the degree of a vertex, , and it is denoted by . If , then is called a pendent vertex or a leaf. A graph of order is called an -vertex graph. Denote by and the -vertex path graph and the -vertex star graph, respectively. The class of all -vertex trees is denoted by . For the (chemical) graph theoretical notation and terminology that are not defined in this paper, refer to [1, 2].
One of the fundamental ideas in CGT (chemical graph theory) is molecular connectivity. Chemical behavior of a compound is dependent upon its structure. QSPR/QSAR (quantitative structure-property/activity relationship) studies are progressive fields of chemical research that focus on the behavior of this dependency. The quantitative relationships are mathematical models that either enable the prediction of a continuous variable (e.g., boiling point and toxicity) or the classification of a discrete variable (e.g., sweet/bitter and toxic/nontoxic) from structural parameters. Actually, CGT has provided many topological indices that have been and are being used in QSPR/QSAR studies for predicting the physicochemical properties of chemical compounds. Topological indices are those graph invariants that found some applications in chemistry [3–6]. For further details about the topological indices and their applications, refer to [3, 7–11] and the references therein.
Molecules can be modeled using graphs in which vertices correspond to atoms of the considered molecules, and the edges correspond to the covalent bonds between atoms . To model the heteroatom molecules, it is better to use the vertex-weighted graphs, which are the graphs whose one or more vertices are distinguished in some way from the rest of the vertices . Let be a vertex-weighted graph with the vertex set , and let be the weight of the vertex for . The augmented vertex-adjacency matrix of is an matrix denoted by and is defined as , where
We associate this index’s name with its inventor Randić by calling it as the variable Randić index. This index was actually introduced within the QSPR/QSAR studies of heteroatom molecules. If is the molecular graph of a homoatomic molecule, then (say), and hence, the variable Randić index becomes
In the rest of this paper, we denote this index by instead of . Clearly, if we take , then the invariant is the classical Randić index [10, 15]. Liu and Zhong  showed that the variable Randić index has more flexibility in characterizing polymers, which can lead to simpler correlations with better correlative accuracy. Details about the chemical applications of the variable Randić index can be found in [5, 7, 9, 12, 15–22] and related references listed therein. It needs to be mentioned here that the variable Randić index seems to have more chemical applications than the several well-known variable indices, for example, the indices considered in [23–33]. However, to the best of the authors’ knowledge, mathematical properties of the variable Randić index, for , have not yet been discussed explicitly in any paper. The main purpose of the present paper is to fill this gap by studying this topological index in a mathematical point of view. Since the trees (that are the connected graphs without cycles) form an important class of graphs both in chemical graph theory as well as in general graph theory, in this paper, we study an extremal problem related to the variable Randić index of the class of trees. We prove that the star graph has the minimum variable Randić index among all trees of a fixed order , where .
2. Main Result
To establish the main result, we prove a lemma first.
Lemma 1. If and , then function defined asis positive-valued.
Proof. We note that the function is strictly increasing in on the interval becausewhere the last inequality holds becauseAlso, note that the value of the function at is 0, which implies that the function is positive-valued for , and hence, function is strictly increasing in on the interval . Due to the identity , function is strictly increasing also in on the interval . It holds that , and hence, for all and .
Transformation 1. For , let be a tree of order containing at least two nonpendent vertices. Let be a vertex of maximum degree, and let , where for . Also, take . Note that . We transform into another tree by removing the edges and adding the edges .
Lemma 2. Let and be the trees defined in Transformation 1. For , it holds that
Proof. By using the definition of the variable Randić index, one haswhereEquation (8) givesSince the vertex has the maximum degree, that is, for every and for , thus, and . Hence, equation (10) yieldsBy using Lemma 1 in (11), we get , as desired.
Next result is a direct consequence of Lemma 2.
Theorem 1. For and , among all trees of a fixed order , star graph is the unique tree with minimum variable Randić index , which isFor and small values of , we calculate the variable Randić index of the trees of order and find that the value of this index does not exceed fromThis suggests the following conjecture.
Conjecture 1. For and , among all trees of a fixed order , path graph is the unique tree with maximum variable Randić index , which is
There are no data concerning the present study except those presented in this manuscript.
Conflicts of Interest
The authors declare that there are no conflicts of interest.
This work was supported by the National University of Computer and Emerging Sciences, Lahore, Pakistan.
J. L. Gross and J. Yellen, Graph Theory, CRC Press, Boca Raton, FL, USA, 2000.
F. Harary, Graph Theory, Addison-Wesley, Toronto, Canada, 1969.
I. Gutman and O. E. Polansky, Mathematical Concepts in Organic Chemistry, Springer-Verlag, New York, NY, USA, 1986.
L. B. Kier and L. H. Hall, Molecular Connectivity in Structure-Activity Analysis, John Wiley and Sons Inc., New York, NY, USA, 1986.
N. Trinajstić, Chemical Graph Theory, CRC Press, Boca Raton, FL, USA, 1992.
M. Randić, “High quality structure-property regressions. Boiling points of smaller alkanes,” New Journal of Chemistry, vol. 24, no. 3, pp. 165–171, 2000.View at: Google Scholar
M. Randić, M. Pompe, D. Mills, and S. C. Basak, “Variable connectivity index as a tool for modeling structure-property relationships,” Molecules (Basel, Switzerland), vol. 9, no. 12, pp. 1177–1193, 2004.View at: Google Scholar
M. Randić, “On computation of optimal parameters for multivariate analysis of structure-property relationship,” Journal of Computational Chemistry, vol. 12, no. 8, pp. 970–980, 1992.View at: Google Scholar
M. Randić, “On history of the Randić index and emerging hostility toward chemical graph theory,” MATCH Communications in Mathematical and in Computer Chemistry, vol. 59, no. 1, pp. 5–124, 2008.View at: Google Scholar
J. C. Dearden, “The use of topological indices in QSAR and QSPR modeling,” in Advances in QSAR Modeling, K. Roy, Ed., pp. 57–88, Springer, Berlin, Germany, 2017.View at: Google Scholar
R. G. Domenech, J. Gálvez, J. V. de Julián-Ortiz, and L. Pogliani, “Some new trends in chemical graph theory,” Chemical Reviews, vol. 108, pp. 1127–1169, 2008.View at: Google Scholar
M. Pompe, “Variable connectivity index as a tool for solving the anti connectivityproblem,” Chemical Physics Letters, vol. 404, pp. 296–299, 2004.View at: Google Scholar
S. Yousaf, A. A. Bhatti, and A. Ali, “On the minimum variable connectivity index of unicyclic graphs with a given order,” Discrete Dynamics in Nature and Society.View at: Google Scholar
A. Behtoei, “Some relations and bounds for the general first zagreb index,” MATCH Communications in Mathematical and in Computer Chemistry, vol. 81, no. 2, pp. 361–370, 2019.View at: Google Scholar
D. Dimitrov and A. Ali, “On the extremal graphs with respect to the variable sum exdeg index,” Discrete Mathematics Letters, vol. 1, pp. 42–48, 2019.View at: Google Scholar
S. Elumalai and T. Mansour, “On the general zeroth-order Randić index of bargraphs,” Discrete Mathematics Letters, vol. 2, pp. 6–9, 2019.View at: Google Scholar
F. Hayat, “On generalized atom-bond connectivity index of cacti,” Iranian Journal of Mathematical Chemistry, vol. 10, pp. 319–330, 2019.View at: Google Scholar
A. Martínez-Pérez and J. M. Rodríguez, “A unified approach to bounds for topological indices on trees and applications,” MATCH Communications in Mathematical and in Computer Chemistry, vol. 82, pp. 679–698, 2019.View at: Google Scholar