Graph Eigenvalues and Projection Constants

Let $\lambda_1(G)\ge \lambda_2(G)\ge \cdots \ge \lambda_n(G)$ 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, $$\lambda_k(G)\le \frac{\lambda_{\mathbb{R}}(k-1)}{2(k-1)}\,n-1,$$ where $$\lambda_{\mathbb{R}}(r)=\sup_{N\ge r}\frac1N \max_{Q\in \mathcal P_r(N)}\sum_{i,j=1}^N |q_{ij}|$$ and $\mathcal P_r(N)$ denotes the set of rank-$r$ orthogonal projections in $\mathbb{R}^{N\times N}$. In Banach space theory, $\lambda_{\mathbb{R}}(r)$ is well known as the maximal absolute projection constant, which has been shown to equal the quasimaximal absolute projection constant $\mu_{\mathbb{R}}(r)$. This yields a new conceptual connection: universal upper bounds on $\lambda_k(G)$ are controlled by the real maximal absolute projection constant $\lambda_{\mathbb{R}}(k-1)$. In dimensions where $\lambda_{\mathbb{R}}(k-1)$ is known explicitly, this gives explicit coefficients. In particular, for $k=3$ this recovers Tang's recent sharp bound $\lambda_3(G)\le n/3-1$. For $k=4$, using $\lambda_{\mathbb{R}}(3)=\frac{1+\sqrt5}{2}$ together with Linz's closed blowups of the icosahedral graph, we obtain the result $$\lambda_4(G) \leq \frac{1+\sqrt5}{12}n-1.$$ The method allows us to transfer known upper bounds on $\lambda_{\mathbb{R}}(k-1)$ to match the best known upper bounds on $\lambda_k(G)$ for other values of $k$, such as $k=5$.