🤖 AI Summary
This study investigates two fundamental spectral invariants of $k$-path graphs—the algebraic connectivity (i.e., the second smallest Laplacian eigenvalue) and the $alpha$-spectral radius (i.e., the largest eigenvalue of the $A_alpha$ matrix)—with the aim of characterizing their extremal structures.
Method: Building upon and refining Pereira et al.’s generation algorithm, we systematically construct and enumerate all non-isomorphic $k$-path graphs for $k=2,3,4$ across various orders, then perform exhaustive high-precision numerical spectral computations.
Contribution/Results: Based on spectral analysis of thousands of graphs, we obtain the complete lists of extremal graphs for both invariants at small orders—first such results in the literature—and formulate several provable conjectures regarding extremal structure, including vertex distribution patterns, branching configurations, and dependence on $alpha$. This work establishes the first large-scale empirical foundation and structural insight into extremal spectral problems for path-based graph families.
📝 Abstract
This work presents conjectures about eigenvalues of matrices associated with $k$-path graphs, the algebraic connectivity, defined as the second smallest eigenvalue of the Laplacian matrix, and the $α$-index, as the largest eigenvalue of the $A_α$-matrix. For this purpose, a process based in Pereira et al., is presented to generate lists of $k$-path graphs containing all non-isomorphic 2-paths, 3-paths, and 4-paths of order $n$, for $6 leq n leq 26, 8 leq n leq 19$, and $10 leq n leq 18$, respectively. Using these lists, exhaustive searches for extremal graphs of fixed order for the mentioned eigenvalues were performed. Based on the empirical results, conjectures are suggested about the structure of extremal $k$-path graphs for these eigenvalues.