Signless Laplacian spectral conditions: Forbidden $4$-cycle and star embeddings
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866908786456788992 |
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| author | Wei, Zhe Lou, Zhenzhen He, Changxiang |
| author_facet | Wei, Zhe Lou, Zhenzhen He, Changxiang |
| contents | The signless Laplacian spectral radius has emerged as a crucial spectral parameter in network science. This paper establishes new extremal results in spectral graph theory by investigating the signless Laplacian spectral radius ($Q$-index) of graphs with forbidden subgraphs. We present a $Q$-spectral analog of classical Nosal-type theorems, providing sharp conditions that guarantee the existence of either a $4$-cycle or a large star $K_{1,m-k}$ in a graph. The main theorem states that for integers $k \geq 0$ and graphs $G$ with size $m \geq \max\{7k+31, k^2+8(k+1)\}$, if $q(G) \geq q(S^+_{m,k+1})$, then $G$ must contain a $4$-cycle or $K_{1,m-k}$, unless $G$ is isomorphic to the extremal graph $S^+_{m,k+1}$ formed by adding $k+1$ independent edges to the star $K_{1,m-k-1}$. This result refines previous work on star embeddings by Wang and Guo [Journal of Algebraic Combinatorics, 59 (2024) 213--224], and completes the $Q$-spectral counterpart to Wang's adjacency spectral theorem for $4$-cycle containment [Discrete Math., 345 (2022) 112973]. Our analysis reveals new insights into how signless Laplacian eigenvalues encode graph structure, with tight bounds demonstrated through explicit extremal graph constructions and asymptotic analysis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_17726 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Signless Laplacian spectral conditions: Forbidden $4$-cycle and star embeddings Wei, Zhe Lou, Zhenzhen He, Changxiang Combinatorics The signless Laplacian spectral radius has emerged as a crucial spectral parameter in network science. This paper establishes new extremal results in spectral graph theory by investigating the signless Laplacian spectral radius ($Q$-index) of graphs with forbidden subgraphs. We present a $Q$-spectral analog of classical Nosal-type theorems, providing sharp conditions that guarantee the existence of either a $4$-cycle or a large star $K_{1,m-k}$ in a graph. The main theorem states that for integers $k \geq 0$ and graphs $G$ with size $m \geq \max\{7k+31, k^2+8(k+1)\}$, if $q(G) \geq q(S^+_{m,k+1})$, then $G$ must contain a $4$-cycle or $K_{1,m-k}$, unless $G$ is isomorphic to the extremal graph $S^+_{m,k+1}$ formed by adding $k+1$ independent edges to the star $K_{1,m-k-1}$. This result refines previous work on star embeddings by Wang and Guo [Journal of Algebraic Combinatorics, 59 (2024) 213--224], and completes the $Q$-spectral counterpart to Wang's adjacency spectral theorem for $4$-cycle containment [Discrete Math., 345 (2022) 112973]. Our analysis reveals new insights into how signless Laplacian eigenvalues encode graph structure, with tight bounds demonstrated through explicit extremal graph constructions and asymptotic analysis. |
| title | Signless Laplacian spectral conditions: Forbidden $4$-cycle and star embeddings |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2601.17726 |