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

Abstract

Regarding a simple graph $ \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_{ \Gamma}(v_{i}) $ and $ w(e) = d_{ \Gamma}(u)+d_{ \Gamma}(v)-2 $, where $ d_{ \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}(\Gamma) $ of a graph $ \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_{ \Gamma}(v_{i})+d_{ \Gamma}(v_{j})-2 $; otherwise, it equals 0. The eigenvalues $ \lambda_{1}^{w}\ge \lambda_{2}^{w}\ge\ldots\ge \lambda_{\nu}^{w} $ of $ A_w $ are called the edge-weighted eigenvalues of $ \Gamma $. We investigate the mathematical properties of $ A_{w}(\Gamma) $'s spectral radius $ \lambda_{1}^{w} $ and energy $ E_{w}(\Gamma) = \sum_{i = 1}^{\nu}|\lambda_{i}^{w}| $. Sharp lower and upper bounds are obtained for $ \lambda_{1}^{w} $ and $ E_{w}(\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.

Keywords

• edge weight energy
• spectral radius
• benzenoid hydrocarbons
• structure-property modeling
• graphical descriptor