Graph Eigenvalues and Projection Constants

Abstract: Let $λ1(G)\ge λ_2(G)\ge \cdots \ge λ_n(G)$ 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 $λk(G)$ 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 $$ λ4(G) \leq \frac{1+\sqrt5}{12}n-1. $$ 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 $k=5$.

• The paper establishes a novel connection between graph eigenvalue bounds and Banach space projection constants via Ky Fan’s variational principle.
• It derives universal sharp bounds for the k-th adjacency eigenvalue, confirming tight results in known cases and improving upon previous methods.
• The findings unify disparate results in spectral graph theory and highlight that advancements in projection constant computation can impact future eigenvalue estimations.

Summary of "Graph Eigenvalues and Projection Constants" (2603.29280)

Introduction and Motivation

The paper addresses extremal spectral graph theory, specifically the optimal upper bounds for the $k$-th adjacency eigenvalue $\lambda_k(G)$ of a graph $G$ of order $n$, where $k \geq 2$. Classical constructions (e.g., unions of cliques) provide lower bounds for $\lambda_k(G)$, but the determination of sharp universal upper bounds for larger $k$ has been elusive, with known results for $k = 1, 2, 3$ and partial progress for $k \geq 4$.

The author introduces a novel framework connecting graph eigenvalue extremal problems to Banach space theory via projection constants, leveraging the variational characterization given by Ky Fan’s minimum principle. The paper demonstrates that universal tight upper bounds for $\lambda_k(G)$ are governed by the maximal absolute projection constant $\lambda_k(G)$0 for $\lambda_k(G)$1.

Main Results

Universal Eigenvalue Bounds via Projection Constants

Let $\lambda_k(G)$2 denote the eigenvalues of the adjacency matrix of $\lambda_k(G)$3. The central result is:

$\lambda_k(G)$4

where $\lambda_k(G)$5 is the supremum over normalized absolute sum of entries in rank-$\lambda_k(G)$6 orthogonal projections, equivalently the maximal absolute projection constant.

For explicit values:

• $\lambda_k(G)$7: $\lambda_k(G)$8 (matching Tang's sharp bound and confirming $\lambda_k(G)$9)
• $G$0: Using $G$1, the bound is $G$2, which Linz demonstrated to be tight via blowups of the icosahedral graph.

Variational and Structural Mechanism

The derivation employs the Ky Fan minimum principle for symmetric matrices and quantifies the smallest sum of $G$3 eigenvalues via the trace against rank-$G$4 projections. The maximal negativity possible for these projections, normalized by graph order, links to the projection constant framework in Banach spaces.

The crucial identification is that, after maximizing over dimensions, the quasimaximal projection constant $G$5 coincides with maximal $G$6 [DeregowskaLewandowska2024], enabling direct translation of Banach-space extremal results to spectral bounds.

Extensions and Comparison with Prior Bounds

The new bound matches and, in explicit cases, improves previous results by Nikiforov [Nikiforov2015], Tang [Tang2026], and Sivashankar [sivashankar2026]. For $G$7, the approach reduces the eigenvalue bound to the unresolved computation of $G$8, with existing constructions (e.g., Paley graphs) verifying the tightness of lower bounds.

The paper subsumes recent results using different machinery (e.g., Gegenbauer polynomials) as special cases of this projection constant approach and clarifies the structural mechanisms leading to equality for key graph families (e.g., the $G$9 family for $n$0 [leonida2025graphs]).

Numerical and Structural Claims

The bounds are supported by explicit calculations of projection constants in small dimensions, previously determined via tight frame constructions and Banach space geometry:

• $n$1 (Grünbaum's conjecture),
• $n$2,
• $n$3.

The paper asserts that these projection constant-based bounds are, in presently known dimensions, provably tight and that the structural extremal graphs attaining equality are characterized by known constructions (e.g., union of cliques for $n$4, Linz's blowups for $n$5).

Implications and Future Directions

Practical Implications

This explicit connection between graph eigenvalues and Banach-space projection constants reframes the spectral extremal problem. It reduces the search for sharp eigenvalue bounds to maximal projection constant computations, which are well-studied in analysis and geometry. Therefore, advances in Banach-space theory can translate directly to improvements in spectral graph theory.

Theoretical Implications

The identification of projection constants as governing parameters for spectral extremal graph theory enables new variational mechanisms and reveals structural links between tight frames, orthogonal projections, and eigenvalue distributions. The approach unifies previous disparate results and provides a clean analytic route to sharp eigenvalue bounds, bypassing ad hoc combinatorial constructions.

Open Problems and Speculation

Computing projection constants for higher dimensions, particularly $n$6 and beyond, remains open and critical for resolving sharp bounds for $n$7 and beyond. The paper suggests that further exploration in extremal tight frames and Banach-space geometry may yield exact eigenvalue bounds for larger $n$8.

As spectral graph theory is central to numerous applications in combinatorics, theoretical computer science, and network analysis, tighter eigenvalue bounds have implications for robustness, expansion, and optimization in these fields.

Conclusion

The paper establishes a direct, explicit correspondence between maximal absolute projection constants in Banach spaces and universal bounds for the $n$9-th adjacency eigenvalue of graphs. The approach yields sharp bounds in known cases and reframes spectral extremal problems in terms of classical geometric analysis. Future advances in projection constant computation will directly impact spectral graph theory, offering a unified analytical lens for extremal eigenvalues and advancing theoretical understanding across mathematics and computer science.