Spectral analysis of the path signless laplacian matrix with QSPR applications to bioactive polyphenols

For a graph with vertex set V( )=v₁,v₂,… ,vₙ , the path matrix P( ) is defined by assigning to its ( h , l )-entry the maximum number of vertex-disjoint paths between vₕ and vₗ whenever h≠ l , and zero otherwise. Based on this matrix, we study the path signless Laplacian matrix PSL( ) and its associated spectral properties. In this paper, we establish several bounds for the path signless Laplacian energy PSLE( ) and investigate the spectral behaviour of PSL( ) under various graph operations, including p -splitting and p -duplicate graphs, together with certain other graph classes. For these graph constructions, we derive the characteristic polynomials, determine the corresponding eigenvalues, analyze the inertia, and compute the resulting path signless Laplacian energies. Furthermore, to demonstrate the applicability of the proposed spectral descriptor, a QSPR analysis of bioactive polyphenols is carried out using the path signless Laplacian energy. Linear regression models are developed to study the correlation between PSLE( ) and several physicochemical properties, including molecular complexity, molecular weight, polar surface area, and boiling point. The obtained results reveal strong correlations, indicating that the path signless Laplacian energy serves as an effective molecular descriptor in chemical graph theory and QSPR modeling.