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Theoretical study demonstrates exact closed formulas for resolvent indices in path and cycle graphs, highlighting novel algebraic tools for structure-activity relationships.
Resolvent based indices associated to a chemical graph are indices defined in terms of the spectrum of the resolvent of the adjacency or a Laplacian-type matrix of a graph. They are closely related to moments of a chemical graph and have potential in analyzing structure activity relationships. In this paper, we derive closed formulas (expressed in terms of the Vieta-Fibonacci and Vieta-Lucas polynomials) for several resolvent based chemical indices for path and cycle graphs that serve as building blocks of many complex chemical graphs. We also pose a conjecture on the asymptotic behavior of Vieta-Fibonacci and Vieta-Lucas polynomials, based on numerical computations and properties of the resolvent based indices.
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Odžak et al. (2026) studied this question.
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