Leap Architecture Matrices: A graph-theoretic framework for unveiling second-neighbor organization in regulatory...

Second-neighbor interactions play a fundamental role in determining the higher-order organization of molecular graphs, yet they are not explicitly represented by conventional degree, distance or adjacency-based graph descriptors. This work introduces the Leap Architecture Matrix (LAM), a novel matrix representation that characterizes the distribution of graph edges between distinct leap-degree classes, thereby preserving the structural organization of second-neighbor connectivity within molecular networks. Based on this representation, three complementary descriptors are developed: the Transition Occupancy Ratio (TOR) which measures the occupancy of leap-transition classes; the Architecture Energy (AE) defined as the sum of the absolute eigenvalues of the Leap Architecture Matrix and the corresponding Spectral Radius (ρ) which quantifies the dominant architectural connectivity. The proposed framework is applied to a dataset of biologically important regulatory amino acids exhibiting diverse structural and physicochemical characteristics. Its discriminative capability is evaluated through comparisons with established molecular descriptors including the Wiener, Randic and first Zagreb indices, together with Pearson correlation analysis, principal component analysis and sensitivity analysis. The results demonstrate that the proposed descriptors capture complementary structural information associated with higher-order molecular organization while maintaining low computational complexity. In particular, the spectral descriptors derived from the Leap Architecture Matrix effectively distinguish subtle variations in molecular architecture that are not fully characterized by conventional topological indices. The proposed matrix framework provides an interpretable and computationally efficient approach for molecular structural characterization and offers a new family of graph-theoretical descriptors with potential applications in molecular similarity analysis, structural classification, spectral graph theory, QSAR/QSPR modeling and computational molecular informatics.