Generalized Sombor index of graphs with given degree sequences: Extremal results and applications

초록

The natural extension of the general sum-connectivity index and general Sombor index for a graph G is termed the generalized Sombor index, defined as: GSα,β(G)=∑νiνj∈E(G)(diβ+djβ)α, where α,β are non-zero real numbers and di is the degree of the vertex νi in G. Let Γ(π) be the class of connected graphs of order n with given degree sequence π. In this paper, under different values of α and β, we show that: (i) The generalized Sombor index exhibits escalating or de-escalating properties; (ii) For any given degree sequence π with minimum degree 1, there exists a BFS-graph that maximizes or minimizes the GSα,β(G) index within Γ(π); (iii) For any c-cyclic degree sequence π with minimum degree 1 and c∈{0,1,2}, an extremal BFS-graph exists in Γ(π) for the index GSα,β(G), and we establish the majorization theorem for such sequences. These results extend the main findings of Damnjanović et al. (2023) and Wei et al. (2023a), as well as prior work on the general sum-connectivity index by Zhang et al. (2017), Liu et al. (2019), and Wei et al. (2023b). Moreover, we apply our results to characterize extremal graphs and trees with respect to key graph parameters. Finally, we demonstrate the chemical predictive power of the GSα,β index over the range −10≤α,β≤10 through QSPR modeling of octane isomer properties, establishing its utility in chemical graph theory.

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