Graph Eigenvalues and Projection Constants
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917372606021632 |
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| author | Wakhare, Tanay |
| author_facet | Wakhare, Tanay |
| contents | Let $λ_1(G)\ge λ_2(G)\ge \cdots \ge λ_n(G)$ denote the adjacency eigenvalues of a graph $G$ of order $n$. We prove that for every $k\geq 2$ and every graph $G$ on $n\geq k$ vertices, $$ λ_k(G)\le \frac{λ_{\mathbb{R}}(k-1)}{2(k-1)}\,n-1, $$ where $$ λ_{\mathbb{R}}(r)=\sup_{N\ge r}\frac1N \max_{Q\in \mathcal P_r(N)}\sum_{i,j=1}^N |q_{ij}| $$ and $\mathcal P_r(N)$ denotes the set of rank-$r$ orthogonal projections in $\mathbb{R}^{N\times N}$. In Banach space theory, $λ_{\mathbb{R}}(r)$ is well known as the maximal absolute projection constant, which has been shown to equal the quasimaximal absolute projection constant $μ_{\mathbb{R}}(r)$.
This yields a new conceptual connection: universal upper bounds on $λ_k(G)$ are controlled by the real maximal absolute projection constant $λ_{\mathbb{R}}(k-1)$. In dimensions where $λ_{\mathbb{R}}(k-1)$ is known explicitly, this gives explicit coefficients. In particular, for $k=3$ this recovers Tang's recent sharp bound $λ_3(G)\le n/3-1$. For $k=4$, using $λ_{\mathbb{R}}(3)=\frac{1+\sqrt5}{2}$ together with Linz's closed blowups of the icosahedral graph, we obtain the result $$ λ_4(G) \leq \frac{1+\sqrt5}{12}n-1. $$ The method allows us to transfer known upper bounds on $λ_{\mathbb{R}}(k-1)$ to match the best known upper bounds on $λ_k(G)$ for other values of $k$, such as $k=5$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_29280 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Graph Eigenvalues and Projection Constants Wakhare, Tanay Combinatorics 05C50, 05C35, 15A42, 46B20 Let $λ_1(G)\ge λ_2(G)\ge \cdots \ge λ_n(G)$ denote the adjacency eigenvalues of a graph $G$ of order $n$. We prove that for every $k\geq 2$ and every graph $G$ on $n\geq k$ vertices, $$ λ_k(G)\le \frac{λ_{\mathbb{R}}(k-1)}{2(k-1)}\,n-1, $$ where $$ λ_{\mathbb{R}}(r)=\sup_{N\ge r}\frac1N \max_{Q\in \mathcal P_r(N)}\sum_{i,j=1}^N |q_{ij}| $$ and $\mathcal P_r(N)$ denotes the set of rank-$r$ orthogonal projections in $\mathbb{R}^{N\times N}$. In Banach space theory, $λ_{\mathbb{R}}(r)$ is well known as the maximal absolute projection constant, which has been shown to equal the quasimaximal absolute projection constant $μ_{\mathbb{R}}(r)$. This yields a new conceptual connection: universal upper bounds on $λ_k(G)$ are controlled by the real maximal absolute projection constant $λ_{\mathbb{R}}(k-1)$. In dimensions where $λ_{\mathbb{R}}(k-1)$ is known explicitly, this gives explicit coefficients. In particular, for $k=3$ this recovers Tang's recent sharp bound $λ_3(G)\le n/3-1$. For $k=4$, using $λ_{\mathbb{R}}(3)=\frac{1+\sqrt5}{2}$ together with Linz's closed blowups of the icosahedral graph, we obtain the result $$ λ_4(G) \leq \frac{1+\sqrt5}{12}n-1. $$ The method allows us to transfer known upper bounds on $λ_{\mathbb{R}}(k-1)$ to match the best known upper bounds on $λ_k(G)$ for other values of $k$, such as $k=5$. |
| title | Graph Eigenvalues and Projection Constants |
| topic | Combinatorics 05C50, 05C35, 15A42, 46B20 |
| url | https://arxiv.org/abs/2603.29280 |