The largest normalized Laplacian eigenvalue and incidence balancedness of simplicial complexes
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| Format: | Preprint |
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2024
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| _version_ | 1866911065029214208 |
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| author | Song, Yi-min Wu, Hui-Feng Fan, Yi-Zheng |
| author_facet | Song, Yi-min Wu, Hui-Feng Fan, Yi-Zheng |
| contents | Let $K$ be a simplicial complex, and let $Δ_i^{up}(K)$ be the $i$-th up normalized Laplacian of $K$. Horak and Jost showed that the largest eigenvalue of $Δ_i^{up}(K)$ 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 $Δ_i^{up}(K)$ 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 $Δ_i^{up}(K)$, which generalizes the corresponding result in graph case. Finally we gave some classes of infinitely many simplicial complexes $K$ with $Δ_i^{up}(K)$ having an eigenvalue $i+2$ by using wedge, Cartesian product and duplication of motifs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_13791 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The largest normalized Laplacian eigenvalue and incidence balancedness of simplicial complexes Song, Yi-min Wu, Hui-Feng Fan, Yi-Zheng Combinatorics 05E45, 05C65, 47J10, 55U05 Let $K$ be a simplicial complex, and let $Δ_i^{up}(K)$ be the $i$-th up normalized Laplacian of $K$. Horak and Jost showed that the largest eigenvalue of $Δ_i^{up}(K)$ 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 $Δ_i^{up}(K)$ 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 $Δ_i^{up}(K)$, which generalizes the corresponding result in graph case. Finally we gave some classes of infinitely many simplicial complexes $K$ with $Δ_i^{up}(K)$ having an eigenvalue $i+2$ by using wedge, Cartesian product and duplication of motifs. |
| title | The largest normalized Laplacian eigenvalue and incidence balancedness of simplicial complexes |
| topic | Combinatorics 05E45, 05C65, 47J10, 55U05 |
| url | https://arxiv.org/abs/2407.13791 |