The largest normalized Laplacian eigenvalue and incidence balancedness of simplicial complexes
| Autor: | Song, Yi-min, Wu, Hui-Feng, Fan, Yi-Zheng |
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| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | Let $K$ be a simplicial complex, and let $\Delta_i^{up}(K)$ be the $i$-th up normalized Laplacian of $K$. Horak and Jost showed that the largest eigenvalue of $\Delta_i^{up}(K)$ is at most $i+2$, and characterized the equality case by the orientable or non-orientable circuits. In this paper, by using the balancedness of signed graphs, we show that $\Delta_i^{up}(K)$ has an eigenvalue $i+2$ if and only if $K$ has an $(i+1)$-path connected component $K'$ such that the $i$-th signed incidence graph $B_i(K')$ is balanced, which implies Horak and Jost's characterization. We also characterize the multiplicity of $i+2$ as an eigenvalue of $\Delta_i^{up}(K)$, which generalizes the corresponding result in graph case. Finally we gave some classes of infinitely many simplicial complexes $K$ with $\Delta_i^{up}(K)$ having an eigenvalue $i+2$ by using wedge, Cartesian product and duplication of motifs. Comment: arXiv admin note: substantial text overlap with arXiv:2405.19078 |
| Databáze: | arXiv |
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