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 $\alpha$-index, as the largest eigenvalue of the $A_{\alpha}$-matrix. For this purpose, a process based on [Discrete Applied Mathematics 164 (2014) 297-303] 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.
From: Claudia Justel [view email]
[v1]
Wed, 26 Nov 2025 15:53:16 UTC (272 KB)
[v2]
Fri, 3 Apr 2026 14:49:59 UTC (273 KB)
[v3]
Sat, 25 Jul 2026 23:23:00 UTC (293 KB)
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