$k$-path graphs: experiments and conjectures about algebraic connectivity and $α$-index
Rafael L. de Paula, Claudia M. Justel, Carla S. Oliveira, Milena S. Carauba
TL;DR
This paper investigates extremal spectral properties of $k$-path graphs with $k\in\{2,3,4\}$ by exhaustively enumerating all non-isomorphic graphs of bounded order using Pereira et al.'s color-sequence framework. It computes the algebraic connectivity $a(G)$ and the $\alpha$-index $\rho_\alpha(G)$ (and $\lambda_2(A_\alpha)$) across the generated lists and identifies structural patterns: the maximal $a(G)$ is achieved by $K_{k-1} \vee P_{n-k+1}$ and the minimal by $P_n^k$, with analogous extremals for $\rho_\alpha(G)$ and $\lambda_2(A_\alpha)$. The study then formulates two conjectures: (i) the unique maximizer of $a(G)$ is $K_{k-1} \vee P_{n-k+1}$ and the unique minimizer is $P_n^k$, and (ii) the maximal $\rho_\alpha$ occurs at $K_{k-1} \vee P_{n-k+1}$ for all $\alpha$, while the maximal $\lambda_2(A_\alpha)$ is the $k$-weak-generalized-fan. The Appendix provides comprehensive tables supporting these findings, highlighting concrete patterns applicable to outerplanar/planar graphs for $k=2,3$ and extending to $k=4$ where non-planarity appears. These results contribute concrete, data-driven conjectures and a publicly available enumeration framework for further theoretical verification.
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.