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arXiv · 2511.21524

$k$-path graphs: experiments and conjectures about algebraic connectivity and $\alpha$-index

Rafael L. de Paula·Claudia M. Justel·Carla S. Oliveira·Milena S. Carauba
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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.

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Rafael L. de Paula, Claudia M. Justel, Carla S. Oliveira, Milena S. Carauba. 2025-11-26. $k$-path graphs: experiments and conjectures about algebraic connectivity and $\alpha$-index. https://arxiv.org/abs/2511.21524

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