Variants of spectral Turán theorems and eigenvectors of graphs
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866908586439868416 |
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| author | Liu, Lele Ning, Bo |
| author_facet | Liu, Lele Ning, Bo |
| contents | In 2002, Nikiforov proved that for an $n$-vertex graph $G$ with clique number $ω$ and edge number $m$, the spectral radius $λ(G)$ satisfies $λ(G) \leq \sqrt{2(1 - 1/ω) m}$, which confirmed a conjecture implicitly suggested by Edwards and Elphick. In this paper, we prove a local version of spectral Turán inequality, which states that $λ^2(G)\leq 2\sum_{e\in E(G)}\frac{c(e)-1}{c(e)}$, where $c(e)$ is the order of the largest clique containing the edge $e$ in $G$. We also characterize the extremal graphs. We prove that our theorem implies Nikiforov's theorem and give an example to show that the difference of Nikiforov's bound and ours is $Ω(\sqrt{m})$ for some cases. Additionally, we establish a spectral counterpart to Ore's problem (1962) which asks for the maximum size of an $n$-vertex graph such that its complement is connected and does not contain $F$ as a subgraph. Our result leads to a new spectral Turán inequality applicable to graphs with connected complements. Finally, we disprove a conjecture of Gregory, asserting that for a connected $n$-vertex graph $G$ with chromatic number $k\geq 2$ and an independent set $S$, we have \[ \sum_{v\in S} x_v^2 \leq \frac{1}{2} - \frac{k-2}{2\sqrt{(k-2)^2 + 4(k-1)(n-k+1)}}, \] where $x_v$ is the component of the Perron vector of $G$ with respect to the vertex $v$. A modified version of Gregory's conjecture is proposed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_16138 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Variants of spectral Turán theorems and eigenvectors of graphs Liu, Lele Ning, Bo Combinatorics In 2002, Nikiforov proved that for an $n$-vertex graph $G$ with clique number $ω$ and edge number $m$, the spectral radius $λ(G)$ satisfies $λ(G) \leq \sqrt{2(1 - 1/ω) m}$, which confirmed a conjecture implicitly suggested by Edwards and Elphick. In this paper, we prove a local version of spectral Turán inequality, which states that $λ^2(G)\leq 2\sum_{e\in E(G)}\frac{c(e)-1}{c(e)}$, where $c(e)$ is the order of the largest clique containing the edge $e$ in $G$. We also characterize the extremal graphs. We prove that our theorem implies Nikiforov's theorem and give an example to show that the difference of Nikiforov's bound and ours is $Ω(\sqrt{m})$ for some cases. Additionally, we establish a spectral counterpart to Ore's problem (1962) which asks for the maximum size of an $n$-vertex graph such that its complement is connected and does not contain $F$ as a subgraph. Our result leads to a new spectral Turán inequality applicable to graphs with connected complements. Finally, we disprove a conjecture of Gregory, asserting that for a connected $n$-vertex graph $G$ with chromatic number $k\geq 2$ and an independent set $S$, we have \[ \sum_{v\in S} x_v^2 \leq \frac{1}{2} - \frac{k-2}{2\sqrt{(k-2)^2 + 4(k-1)(n-k+1)}}, \] where $x_v$ is the component of the Perron vector of $G$ with respect to the vertex $v$. A modified version of Gregory's conjecture is proposed. |
| title | Variants of spectral Turán theorems and eigenvectors of graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2312.16138 |