$k$-path graphs: experiments and conjectures about algebraic connectivity and $α$-index
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866910098528403456 |
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| author | de Paula, Rafael L. Justel, Claudia M. Oliveira, Carla S. Carauba, Milena S. |
| author_facet | de Paula, Rafael L. Justel, Claudia M. Oliveira, Carla S. Carauba, Milena S. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_21524 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $k$-path graphs: experiments and conjectures about algebraic connectivity and $α$-index de Paula, Rafael L. Justel, Claudia M. Oliveira, Carla S. Carauba, Milena S. Discrete Mathematics Combinatorics 05C50 05C75 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. |
| title | $k$-path graphs: experiments and conjectures about algebraic connectivity and $α$-index |
| topic | Discrete Mathematics Combinatorics 05C50 05C75 |
| url | https://arxiv.org/abs/2511.21524 |