A new variant of the Sombor matrix: bounds on spectral radius and energy with applications to QSPR analysis of COVID-19 drugs

In this paper, we introduce a novel variant of the Sombor matrix, denoted as $\mathcal{NS}o(\zeta)$, for a simple graph $\zeta(\mathcal{V}, \mathcal{E})$. The matrix is defined such that for $i \neq j$, the $(i,j)$-entry is given by $\sqrt{d_{i}^{2}+d_{j}^{2}}$, where $d_i$ represents the degree of the $i^{\text{th}}$ vertex, and zero otherwise. Let $\eta_1 \geq \eta_2 \geq \cdots \geq \eta_{\rho}$ denote the eigenvalues of $\mathcal{NS}o(\zeta)$, with $\eta_1$ being the spectral radius. The Sombor energy $E_{\mathcal{NS}o}(\zeta)$ is defined as the sum of the absolute values of these eigenvalues. We derive upper and lower bounds for both $\eta_1$ and $E_{\mathcal{NS}o}(\zeta)$ in terms of the first Zagreb index ($M_1(\zeta)$). As an application, we perform a Quantitative Structure-Property Relationship (QSPR) analysis using a dataset of drugs employed in the treatment of COVID-19 patients. We construct linear, quadratic, and cubic regression models to explore the relationship between the physicochemical properties of these drugs and their corresponding $E_{\mathcal{NS}o}(\zeta)$ values, with the models visualized through graphical representations.

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