Degree Deviation and Spectral Radius
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866912041248227328 |
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| author | Rautenbach, Dieter Werner, Florian |
| author_facet | Rautenbach, Dieter Werner, Florian |
| contents | For a finite, simple, and undirected graph $G$ with $n$ vertices, $m$ edges, and largest eigenvalue $λ$, Nikiforov introduced the degree deviation of $G$ as $s=\sum_{u\in V(G)}\left|d_G(u)-\frac{2m}{n}\right|$. Contributing to a conjecture of Nikiforov, we show $λ-\frac{2m}{n}\leq \sqrt{\frac{2s}{3}}$. For our result, we show that the largest eigenvalue of a graph that arises from a bipartite graph with $m_{A,B}$ edges by adding $m_A$ edges within one of the two partite sets is at most $\sqrt{m_A+m_{A,B}+\sqrt{m_A^2+2m_Am_{A,B}}}$, which is a common generalization of results due to Stanley and Bhattacharya, Friedland, and Peled. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_14956 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Degree Deviation and Spectral Radius Rautenbach, Dieter Werner, Florian Combinatorics For a finite, simple, and undirected graph $G$ with $n$ vertices, $m$ edges, and largest eigenvalue $λ$, Nikiforov introduced the degree deviation of $G$ as $s=\sum_{u\in V(G)}\left|d_G(u)-\frac{2m}{n}\right|$. Contributing to a conjecture of Nikiforov, we show $λ-\frac{2m}{n}\leq \sqrt{\frac{2s}{3}}$. For our result, we show that the largest eigenvalue of a graph that arises from a bipartite graph with $m_{A,B}$ edges by adding $m_A$ edges within one of the two partite sets is at most $\sqrt{m_A+m_{A,B}+\sqrt{m_A^2+2m_Am_{A,B}}}$, which is a common generalization of results due to Stanley and Bhattacharya, Friedland, and Peled. |
| title | Degree Deviation and Spectral Radius |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2409.14956 |