Generalized spectral Turán problems for disjoint cliques

Abstract: The generalized Turán number $\text{ex}(n, H, F)$ denotes the maximum number of copies of $H$ in an $n$-vertex $F$-free graph. Let $kK_{r+1}$ be the disjoint union of $k$ copies of the complete graph $K_{r+1}$. Recently, Gerbner determined $\text{ex}(n, K_{t},kK_{r+1})$ for all sufficiently large $n$. In this paper, we study a spectral analogue of this problem via the $t$-clique tensor of a graph. We prove that if an $n$-vertex $kK_{r+1}$-free graph $G$ maximizes the $t$-clique spectral radius, then for sufficiently large $n$, $G$ is the join of a complete graph $K_{k-1}$ and the $r$-partite Turán graph $T_{r}(n-k+1)$. This establishes a spectral counterpart of Gerbner's Theorem. Moreover, in the case $t=2$, our result recovers a theorem of Ni, Wang, and Kang on the maximum spectral radius of $kK_{r+1}$-free graphs.

• The paper establishes that the extremal kK₍r₊₁₎‐free graph maximizing the t‐clique spectral radius is uniquely the join of K₍k₋₁₎ and Tᵣ(n-k+1).
• It leverages advanced spectral tensor analysis, including the generalized Perron–Frobenius theorem, combined with stability and combinatorial techniques.
• The results bridge classical Turán problems with spectral extremal graph theory, setting a foundation for further generalizations to hypergraph and non-clique structures.

Spectral Generalizations of Turán Problems for Disjoint Cliques

Introduction and Foundations

This work addresses a spectral analogue of generalized Turán-type extremal problems, specifically for forbidden families of disjoint cliques. Given positive integers $k$, $r$, and $t$ with $r \geq t \geq 2$, the classical Turán problem determines the maximal edge count or $t$-clique count in an $n$-vertex graph excluding $k$ vertex-disjoint $(r+1)$-cliques, $kK_{r+1}$. The generalized Turán number $\mathrm{ex}(n, K_t, kK_{r+1})$ denotes the maximum number of $r$0 copies in a $r$1-free $r$2-vertex graph. Prior results, notably due to Gerbner, settled the Turán number for large $r$3 by showing that the unique extremal construction is the join of $r$4 and the $r$5-partite Turán graph $r$6.

The central innovation here is to formulate and resolve the spectral generalization—maximizing the $r$7-clique spectral radius, as defined via the $r$8-clique tensor—among all $r$9-vertex, $t$0-free graphs. The $t$1-clique spectral radius is the largest eigenvalue of the tensor encoding the adjacency structure of $t$2-cliques. This unifies and extends spectral generalizations for forbidden cliques and overlaps directly with existing results when $t$3 (the classical adjacency spectral radius).

Key Results and Technical Contributions

The authors establish that for all $t$4 sufficiently large (parameter bounds are explicit but omitted here for brevity), the extremal $t$5-free graph maximizing the $t$6-clique spectral radius is uniquely the join $t$7. This result constitutes a spectral counterpart to the generalized Turán theorem for disjoint cliques. In the $t$8 case, the result recovers and subsumes earlier statements about the ordinary adjacency spectral radius for forbidden disjoint cliques, establishing the extremality of $t$9 for both edge count and spectral radius.

From a technical standpoint, the proof combines elaborate structural stability results à la Simonovits, spectral tensor analysis using the Perron–Frobenius theorem generalized to hypermatrices, and combinatorial arguments to localize high spectral mass. An important aspect is the extension and exploitation of the $r \geq t \geq 2$0-clique tensor framework, originally developed by Cooper–Dutle and Liu–Bu, and its alignment with combinatorial and spectral stability tools in extremal graph theory.

A summary of the key theorem:

• Main Theorem: For $r \geq t \geq 2$1, $r \geq t \geq 2$2, and $r \geq t \geq 2$3 large, if $r \geq t \geq 2$4 is a $r \geq t \geq 2$5-free graph, then

$r \geq t \geq 2$6

with equality only for the extremal construction.

Notably, the proof yields structural information: any putative extremal graph must, up to $r \geq t \geq 2$7 edge modifications, be the join of a clique and a nearly balanced $r \geq t \geq 2$8-partite graph, and indeed all exceptional vertices are absorbed into the clique or the partite structure.

Methodologies and Structural Analysis

The core methodology integrates the following components:

1. Spectral Tensor Theory: The $r \geq t \geq 2$9-clique tensor captures all $t$0-cliques as order-$t$1 hypermatrix entries. The spectral radius of the tensor encodes extremal $t$2-clique density in a manner analogous to the adjacency spectral radius capturing edge extremality. The Perron–Frobenius theorem (extended for weakly irreducible, nonnegative symmetric tensors) guarantees a positive eigenvector, instrumental for stability and monotonicity arguments.
2. Stability and Partitioning: Application of stability theorems for spectral extremal graphs enables reduction to near-partite structure, with exceptional sets managed through careful counting and stability inequalities.
3. Iterative Refinement and Localization: The argument iteratively sharpens the possible structural deviations, controlling high-degree vertices and localizing where large eigenvector coordinates can occur. At each stage, possible deviations are shown to yield either a smaller spectral value or create forbidden substructures, ruling them out.

Additionally, the proof leverages refined inequalities relating spectral radius and clique counts, and uses matching-theoretic results (cf. Chvátal-Hanson) to bound possible in-partite edges given the $t$3-free restriction.

Implications and Future Directions

The results solidify the tight connection between combinatorial and spectral extremal graph theory in the regime of forbidden disjoint cliques, answering open analogues about maximal tensor spectral radii and their achieving graphs. The approach demonstrates the efficacy of combining tensor-based spectral graph theory with stability analysis, suggesting potential in extending to broader forbidden subgraph configurations or even more general hypergraph settings.

On a theoretical level, the paper lays groundwork for further investigation into generalized spectral extremal problems:

• Exploring spectral analogues for other subgraph configurations (beyond cliques or their unions).
• Further generalizing spectral extremal results to uniform hypergraphs and related tensor structures.
• Investigating algorithmic implications for spectral extremal graph identification in practice.

Moreover, the link between $t$4-clique tensors and extremal subgraph counts may inform spectral threshold phenomena and stability in random and pseudorandom contexts.

Conclusion

This work provides a comprehensive and technically robust resolution of the spectral Turán-type problem for forbidden $t$5 subgraphs via the $t$6-clique tensor framework. It establishes that maximizing the $t$7-clique spectral radius within $t$8-free graphs compels the exact same class of extremal constructions as the classical and combinatorial Turán problem—validating the spectral methodology as a powerful extension of extremal combinatorics and reinforcing the structural correspondence between spectrum and forbidden substructures. The results unify prior spectral extremal results for cliques, demonstrate deep stability phenomena, and set a high standard for further spectral generalizations in extremal graph theory (2604.17242).