Hexagonal and k-hexagonal graph's normalized Laplacian spectrum and applications
Hao Li, Xinyi Chen, Hao Liu
TL;DR
The authors address the problem of determining the completely normalized Laplacian spectrum for hexagonal and k-hexagonal graph families built from an arbitrary simple connected graph $G$, and they derive how the spectrum propagates through iterative construction. Their main approach uses a detailed linear-equation analysis of the transformed graph structure, yielding a recursive spectral map: nontrivial eigenvalues of the previous stage generate five eigenvalues at the next stage via roots of a fixed polynomial, together with explicit fixed eigenvalues whose multiplicities depend on $N_{n-1}$, $E_{n-1}$ and bipartiteness. Building on this spectrum, they obtain closed-form and recurrence formulas for key graph invariants, including Kemeny’s constant, the multiplicative degree-Kirchhoff index, and the number of spanning trees, for both $H_n(G)$ and $H^k_n(G)$, with all expressions tied to the initial data of $G$ and its basic growth parameters. A notable finding is that the same invariant formulas hold for $k\ge2$ and for $k=1$, explained by fixed spectral components that preserve the recurrence structure, and the work suggests extensions to more general edge-substitution constructions such as $P^{k,t}(G)$.
Abstract
Substituting each edge of a simple connected graph $G$ by a path of length 1 and $k$ paths of length 5 generates the $k$-hexagonal graph $H^k(G)$. Iterative graph $H^k_n(G)$ is produced when the preceding constructions are repeated $n$ times. According to the graph structure, we obtain a set of linear equations, and derive the entirely normalized Laplacian spectrum of $H^k_n(G)$ when $k = 1$ and $k \geqslant 2$ respectively by analyzing the structure of the solutions of these linear equations. We find significant formulas to calculate the Kemeny's constant, multiplicative degree-Kirchhoff index and number of spanning trees of $H^k_n(G)$ as applications.