On spectral and structural properties of m -generalized fullerene graphs F ( m, k )

This article investigates the structural and spectral properties of the family of m-generalized fullerene graphs known as elongated barrels, denoted by F (m, k). These graphs serve as important mathematical models for nanotubes with varying width (m) and length (k). Guided by initial computational analysis, we formulate and subsequently provide rigorous mathematical proofs for four key conjectures regarding their spectral properties. Our main results demonstrate that: (1) the algebraic connectivity is a strictly decreasing function of both the length parameter k and the width parameter m; (2) the adjacency spectrum becomes asymptotically symmetric about the origin as the graph length increases; (3) the adjacency matrix is nonsingular for all members of this family; and (4) the smallest adjacency eigenvalue converges to −3 as k → ∞. This research combines computational discovery with analytical proof to offer a precise characterization of the spectral behavior of this important class of molecular graphs.