Abstract
Mathematical modeling or numerical coding of the molecular structures play a significant role in the studies of the quantitative structure-activity relationships (QSAR) and quantitative structures property relationships (QSPR). In 1972, the entire energy of π-electron of a molecular graph is computed by the addition of square of degrees (valencies) of its vertices (nodes). Later on, this computational result was called by the first Zagreb index and became well studied topological index in the field of molecular graph theory. In this paper, for k ∈ N (set of counting numbers), we define four subdivision-related operations of graphs in their generalized form named by S k , R k , Q k and T k . Moreover, using these operations and the concept of the cartesian product of graphs, we construct the generalized F k -sum graphs Γ 1+Fk Γ 2 , where F k ∈ {S k , R k , Q k , T k } and Γ i are any connected graphs for i ∈ {1, 2}. Finally, the first and second Zagreb indices are computed for the generalized F k -sum graphs in terms of their factor graphs. In fact, the obtained results are a general extension of the results Eliasi et al. and Deng et al. who studied these operations for exactly k = 1 and computed the Zagreb indices for only F 1 -sum graphs respectively.
Highlights
Topological indices (TI’s) are graphical invariants that relate a numeric number to a graph which is structurally invariant
For the counting number k ≥ 1, we define the generalised version of the subdivision related operations of graphs presented by the symbols of Sk, Rk, Qk and Tk. Using these operations and the concept of cartesian product of graphs, we construct the generalized Fk -sum graphs 1+Fk 2, where Fk ∈ {Sk, Rk, Qk, Tk } and i are any connected graphs for i ∈ {1, 2}
Main results of the Zagreb indices for the generalized Fk -sum graphs 1 +Sk 2, 1 +Rk 2, 1 +Qk 2 and 1 +Tk 2 are presented
Summary
Topological indices (TI’s) are graphical invariants that relate a numeric number to a graph which is structurally invariant. Gutman and Trinajsti [15] computed the entire energy of π -electron of a molecule by the addition of square of degrees (valances) of the vertices (nodes) of its structure In the literature, this addition is known as first Zagreb index. Liu et al.: Computing Zagreb Indices of the Subdivision-Related Generalized Operations of Graphs presented the operations (S1, R1, Q1, T1) related to the subdivision of at the same time with reference to their edge and vertex sets and obtained the Wiener index of with respect to these operations. For F1 ∈ {S1, R1, Q1, T1}, Eliasi and Taeri [9] defined the F1-sum graph 1 +F1 2 using cartesian product on the graphs F1( 1) and 2 They discussed the Wiener indices of the F1-sum graphs 1 +S1 2, 1 +R1 2, 1 +Q1 2 and 1 +T1 2. The rest of the paper is settled as; Section II consists on the basic definitions and terminologies, Section III includes the main results and Section IV contains the conclusion with the applications of the obtained results
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