Abstract
For a molecular graph Γ, the general sum-connectivity index is defined as χ β (Γ) = Σ vw∈E(Γ) [d Γ (v) + d Γ (w)] β , where β ∈ R and d Γ (v) denotes the degree of the vertex v in the molecular graph Γ. The problem of finding best possible upper and lower bound for certain topological index is of fundamental nature in extremal graph theory. Akhtar and Imran [J. Inequal. Appl. (2016) 241] obtained the sharp bounds of general sum-connectivity index for four graph operations (F-sum graphs) introduced by Eliasi and Taeri [Discrete Appl. Math. 157: 794-803, 2009)]. In this paper, for β ∈ N, we figured out and improved the sharp bounds of the general sum-connectivity index for F-sum graphs, where F ∈ {R, Q, T}. Several examples are presented to elaborate and compare the results of improved bounds with existing sharp bounds. In addition, we obtained exact formula of general sum-connectivity index for F-sum graphs, when F = S.
Highlights
Assume = (V ( ), E( )) be a simple, connected and finite molecular graph
A path graph or linear graph Pn of length n − 1 be a graph consisting of vertex set {vi : i = 1, 2, . . . , n} and edge set {vivi+1 : i = 1, 2, . . . , n − 1}
A cycle Cn having length n be a graph consisting of vertex set {vi : i = 1, 2, . . . , n} and edge set {vivi+1 : i = 1, 2, . . . , n − 1} ∪ {vnv1}
Summary
M. Ahmad et al.: Exact Formula and Improved Bounds index F( ) by setting α = 2 and α = 3 in FGZI, respectively [9], [11]. In [15], Zhou and Trinajstić (2010) presented the concept of general sum-connectivity index (GSCI) and established Nordhaus-Gaddum-type results for GSCI. It is defined as follows: χβ ( ) =. The cartesian product of two simple graphs 1 and 2 is a new graph denoted by 1 2 whose vertex set is V ( 1) V ( 2) = V ( 1) × V ( 2) and whose edge set is the VOLUME 7, 2019. We derived exact formula for GSCI of graph 1+S 2 in terms of certain topological indices of base graphs
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