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.

翻译：本文提出了关于$k$-路径图关联矩阵特征值的猜想，包括代数连通性（定义为拉普拉斯矩阵的第二小特征值）以及$α$指标（定义为$A_α$矩阵的最大特征值）。为此，基于Pereira等人的方法，本文提出了一种生成$k$-路径图列表的方案，分别包含所有非同构的2-路径图、3-路径图和4-路径图，其阶数$n$满足$6 \leq n \leq 26$、$8 \leq n \leq 19$和$10 \leq n \leq 18$。利用这些列表，针对上述特征值，对固定阶数的极值图进行了穷举搜索。基于实证结果，提出了关于这些特征值对应的极值$k$-路径图结构的猜想。