On the Balaban Index of Chain Graphs
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On the Balaban Index of Chain Graphs Kinkar Chandra Das1 Received: 5 August 2020 / Revised: 8 November 2020 / Accepted: 12 November 2020 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract The Balaban index and sum-Balaban index of a connected (molecular) graph G are defined as J (G) =
m μ+1
S J (G) =
1 and σG (u)σG (v) 1 m √ uv∈E(G) σG (u)+σG (v) , μ+1 uv∈E(G)
√
respectively, where m is the number of edges, μ is the cyclomatic number, σG (u) is the sum of distances between vertex u and all other vertices of G. In this paper, we establish that n n − 1, −1 K (DS(n − 3, 1)) > K (DS(n − 4, 2)) > · · · > K DS 2 2 (K = J , S J ), where DS( p, q) is a double star on n (= p + q + 2, p ≥ q) vertices. As an application, we determine the extremal graphs of the Balaban index and the sum-Balaban index in the class of chain graphs G on n vertices, where G is a tree or a unicyclic graph. Finally, we give an open problem on Balaban (sum-Balaban) index of connected chain graphs. Keywords Molecular graph · Balaban index · Sum-Balaban index AMS Classification: 05C07
Communicated by Xueliang Li.
B 1
Kinkar Chandra Das [email protected] Department of Mathematics, Sungkyunkwan University, Suwon 16419, Republic of Korea
123
K. C. Das
1 Introduction A molecular graph is a simple, connected and undirected graph corresponding to structural formula of a chemical compound, so that vertices or nodes of the graph correspond to atoms of the molecule, and edges or lines of the graph correspond to the bonds between these atoms. Chemical and molecular graphs have several fundamental applications in chemoinformatics, quantitative structure-property relationships (QSPR), quantitative structure-activity relationships (QSAR), virtual screening of chemical libraries, and computational drug design. Chemoinformatics applications of graphs include chemical structure representation and coding, database search and retrieval, and physicochemical property prediction. One of the simplest methods that have been devised for correlating structures with biological activities or physical– chemical properties involves molecular descriptors called topological indices. Since physical properties or bioactivities are expressed in numbers whereas chemical structures are discrete graphs, in order to associate graphs with numbers one has to rely on graph-theoretical invariants such as local vertex invariants, e.g., vertex degree, distance sum, etc. Topological indices (TIs) are digital counterparts of chemical structures and therefore represent these discrete molecular constitutional formulas by numerical functions. Hundreds of topological indices have been introduced so far. With respect to the invariant which plays a crucial role in the definition, we can divide topological indices into three types: degree-based indices, distance-based indices and spectrumbased indices. Degree-based indices [12,17–20] include Zagreb indices, ABC-index, Randi´c index, G A-index, etc. Distance-based indices [6,
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