The characteristic polynomials of $r$-uniform hypercycles with length $l$
Dong Bo, Duan Cunxiang, Wang Ligong
TL;DR
This work determines exact characteristic polynomials for the $r$-uniform hypercycles $C_l^{(r)}$ by leveraging higher-order traces of the adjacency tensor and the BEST Theorem, linking hypergraph spectra to squared eigenvalues of signed subgraphs. The authors derive a general expression for $\phi(C_l^{(r)};\lambda)$ in terms of transformed path polynomials $\tilde{\phi}(P_j;\lambda)$ and coefficients $m_j$ obtained from spectral traces $\mathrm{Tr}_{jr}(\mathcal{A}(C_l^{(r)}))$, with $m_i$ computed from an inverted moment matrix $S$ (or an efficient $B$-based scheme). They provide explicit results for $C_5^{(r)}$ and $C_6^{(r)}$ to illustrate the method and corroborate consistency with established cases ($r=3$) for small $l$. The approach unifies the computation of hypercycle spectra and advances explicit spectral characterizations of structured uniform hypergraphs, enabling direct computation of eigenvalues from the derived polynomials.
Abstract
Let $C_{l}$ be a cycle with length $l.$ The $r$-uniform hypercycle with length $l$ is obtained by adding $r-2$ new vertices in every edge of $C_{l},$ denoted by $C_l^{(r)}$. In this paper, we deduce some higher-order traces for the adjacent tensor of $C_l^{(r)}$ by BEST Theorem. Then we obtain higher-order spectral moments according to the relationship between eigenvalues of power hypergraphs and eigenvalues of signed graphs. Finally, the general expression of the characteristic polynomials of $C_l^{(r)}$ is given. Furthermore, by using this general expression, we present the characteristic polynomials of $C_5^{(r)}$ and $C_6^{(r)}$ as examples.