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

TL;DR

This paper investigates eigenvalues of matrices related to $k$-path graphs, presenting conjectures on their algebraic connectivity and $oldsymbol{ ext{α}}$-index based on exhaustive computational experiments.

Contribution

It introduces a process to generate all non-isomorphic $k$-path graphs for certain sizes and uses exhaustive searches to formulate conjectures on extremal eigenvalue structures.

Findings

Conjectures about extremal $k$-path graphs for algebraic connectivity.

Empirical evidence for eigenvalue behavior in $k$-path graphs.

Generated comprehensive graph lists for sizes up to 26 nodes.

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 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.