A novel edge-weighted matrix of a graph and its spectral properties with potential applications

Regarding a simple graph $\mathrm{\Gamma }$ possessing $\nu$ vertices ( $\nu$-vertex graph) and $m$ edges, the vertex-weight and weight of an edge $e=uv$ are defined as $w(v_i)=d_{\mathrm{\Gamma }}(v_i)$ and $w(e)=d_{\mathrm{\Gamma }}(u)+d_{\mathrm{\Gamma }}(v)-2$, where $d_{\mathrm{\Gamma }}(v)$ is the degree of $v$. This paper puts forward a novel graphical matrix named the edge-weighted adjacency matrix (adjacency of the vertices) $A_w(\mathrm{\Gamma })$ of a graph $\mathrm{\Gamma }$ and is defined in such a way that, for any $v_i$ that is adjacent to $v_j$, its $(i,j)$-entry equals $w(e)=d_{\mathrm{\Gamma }}(v_i)+d_{\mathrm{\Gamma }}(v_j)-2$; otherwise, it equals 0. The eigenvalues ${\lambda }_1^w\ge {\lambda }_2^w\ge \dots \ge {\lambda }_{\nu }^w$ of $A_w$ are called the edge-weighted eigenvalues of $\mathrm{\Gamma }$. We investigate the mathematical properties of $A_w(\mathrm{\Gamma })$'s spectral radius ${\lambda }_1^w$ and energy $E_w(\mathrm{\Gamma })=\sum _{i=1}^{\nu }|{\lambda }_i^w|$. Sharp lower and upper bounds are obtained for ${\lambda }_1^w$ and $E_w(\mathrm{\Gamma })$, and the respective extremal graphs are characterized. Further, we employ these spectral descriptors in structure-property modeling of the physicochemical properties of polycyclic aromatic hydrocarbons for a set of benzenoid hydrocarbons (BHs). Detailed regression analysis showcases that edge-weighted energy outperforms classical adjacency energy in structure-property modeling of the physicochemical properties of BHs.