Novel Roman domination-based graph energies for QSPR analysis of neuroprotective herbal compounds in Alzheimer’s disease treatment

Alzheimer’s disease (AD) is a progressive neurodegenerative disorder for which U.S. Food and Drug Administration (FDA)-approved drugs provide only temporary symptomatic relief and often cause adverse effects. Plant-derived bioactive phytochemicals are emerging as promising alternatives due to their multi-targeted neuroprotective properties and reduced toxicity. In this article, herbal anti-Alzheimer’s compounds are analyzed using a novel graph molecular modeling. In chemical graph theory, molecular structures are represented as isomorphic molecular graphs <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m1"> <mml:mrow> <mml:mi>G</mml:mi> <mml:mrow> <mml:mfenced open="(" close=")" separators="|"> <mml:mrow> <mml:mi>V</mml:mi> <mml:mo>,</mml:mo> <mml:mi>E</mml:mi> </mml:mrow> </mml:mfenced> </mml:mrow> </mml:mrow> </mml:math> , where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m2"> <mml:mrow> <mml:mi>V</mml:mi> </mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m3"> <mml:mrow> <mml:mi>E</mml:mi> </mml:mrow> </mml:math> denote the set of vertices (atoms) and edges (chemical bonds) respectively. Classical graph matrices such as adjacency and Laplacian matrices capture the molecular connectivity but fail to account for hierarchical differences in atomic influence. To address this limitation, Roman domination is employed to represent the hierarchical dominance of atoms within molecular structures. A Roman domination function (RDF) on a graph <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m4"> <mml:mrow> <mml:mi>G</mml:mi> <mml:mrow> <mml:mfenced open="(" close=")" separators="|"> <mml:mrow> <mml:mi>V</mml:mi> <mml:mo>,</mml:mo> <mml:mi>E</mml:mi> </mml:mrow> </mml:mfenced> </mml:mrow> </mml:mrow> </mml:math> is a mapping <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m5"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:mi>V</mml:mi> <mml:mo>→</mml:mo> <mml:mrow> <mml:mfenced open="{" close="}" separators="|"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:mfenced> </mml:mrow> </mml:mrow> </mml:math> such that every atom <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m6"> <mml:mrow> <mml:mi>v</mml:mi> </mml:mrow> </mml:math> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m7"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mrow> <mml:mfenced open="(" close=")" separators="|"> <mml:mrow> <mml:mi>v</mml:mi> </mml:mrow> </mml:mfenced> </mml:mrow> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> has at least one adjacent atom <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m8"> <mml:mrow> <mml:mi>u</mml:mi> </mml:mrow> </mml:math> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m9"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mrow> <mml:mfenced open="(" close=")" separators="|"> <mml:mrow> <mml:mi>u</mml:mi> </mml:mrow> </mml:mfenced> </mml:mrow> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , reflecting the hierarchical dominance within the isomorphic molecular graph. Based on this principle, the Roman domination-based matrices and corresponding graph energies are introduced in this article. Quantitative Structure-Property Relationship (QSPR) graph models are developed using the Roman energies through linear, quadratic and cubic regression analysis. The results demonstrate superior performance compared to classical approaches, with the quadratic regression showing the strongest correlations and lowest standard error. Internal validation through the Y-randomization and Leave-One-Out Cross-Validation methods confirmed the stability of the models, while external validation on the herbal compound Kaempferol ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m10"> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0.993</mml:mn> </mml:mrow> </mml:math> ) further supported their predictive reliability. These findings underscore the robustness of Roman energies, establishing them as powerful molecular descriptors that offer enhanced accuracy in the QSPR analysis and hold promise for applications in drug design, materials informatics and computational chemistry.