The high order spectral extrema of $2K_r$-free graphs

Abstract: In this paper, we determine the graphs with maximum value of the sum number from $k$-clique spectral radius to $(2r-1)$-clique spectral radius among all $2K_{r}$-free graphs on $n$ vertices for $r\le k$ and large $n$. We also determine the graphs with maximum $3$-clique spectral radius among all $2K_{3}$-free graphs on $n$ vertices. Our results are spectral versions of some results on generalized Turán numbers.

• The paper presents spectral Turán-type extremal results for 2K_r-free graphs, identifying unique extremal configurations maximizing sums of clique tensor spectral radii.
• It employs advanced tensor spectral radius inequalities and weak irreducibility conditions to extend classical combinatorial extremal results to higher-order spectra.
• The work bridges combinatorial optimization with hypergraph spectral theory, offering methodologies for detecting forbidden clique substructures in graph analysis.

High Order Spectral Extrema in $2K_r$-Free Graphs

Introduction

This work addresses spectral extremal problems for $2K_r$-free graphs, focusing on higher-order clique tensors and their spectral radii. Extending classical and generalized Turán-type results, the authors develop spectral analogues that provide tight upper bounds for sums of clique tensor spectral radii and identify extremal graphs achieving these bounds. The results efficiently bridge extremal combinatorial analysis with non-linear spectral graph theory, particularly in the context of forbidden-disjoint clique substructures.

Preliminaries: Higher-Order Spectra and Clique Tensors

Let $G$ be a finite, undirected, simple graph. For $r \geq 2$, the $r$-clique tensor $\mathcal{A}_r(G)$ is a symmetric, non-negative, $r$-order, $n$-dimensional tensor, where entries correspond to the existence of $r$-cliques. The $r$-clique spectral radius $2K_r$0 is defined as the maximum modulus of the eigenvalues of $2K_r$1 (in the sense of Qi and Lim). Notably, for $2K_r$2, $2K_r$3 reduces to the usual adjacency matrix.

A $2K_r$4-free graph is a graph without two vertex-disjoint $2K_r$5 subgraphs. This strong restriction creates structural and extremal limitations on clique mixing within the host graph. The generalized Turán number $2K_r$6 counts the maximal number of $2K_r$7 subgraphs appearing in an $2K_r$8-free $2K_r$9-vertex graph, and foundational work by Zykov, Erdős, and later Yuan-Yang and others, has characterized extremal combinatorial structures in various cases.

However, the spectral versions—focusing on extremal values of tensor spectral radii, rather than mere subgraph counts—have only recently become tractable, due to advances in the theory of hypergraph spectra and weak irreducibility of tensors.

Main Results

1. Spectral Turán-Type Extremal Theorems for $G$0-Free Graphs

For $G$1 and sufficiently large $G$2, it is shown that among all $G$3-free graphs, the sum

$G$4

attains its maximum uniquely on the graph $G$5. Here, $G$6 is the balanced complete $G$7-partite graph on $G$8 vertices. The bound is tight: if $G$9, extremality only occurs when the graph contains a single $r \geq 2$0-clique; for $r \geq 2$1, only on the specified $r \geq 2$2-partite graph.

This result is a spectral adaptation of Gerbner and Patkós' generalized Turán-type theorem for intersecting cliques and coincides with previously known combinatorial extremal configurations [Gerbner, Discrete Mathematics 2024; Yuan & Yang, Graphs and Combinatorics 2022].

2. Extremal Graphs for Spectral 3-Clique Radius in $r \geq 2$3-Free Graphs

A detailed analysis is provided for spectral radius extremality with respect to 3-clique tensors in $r \geq 2$4-free graphs. The unique extremal structure depends nontrivially on $r \geq 2$5:

• For $r \geq 2$6: $r \geq 2$7.
• For $r \geq 2$8: $r \geq 2$9 or a graph obtained by joining an isolated vertex to one vertex in $r$0.
• For $r$1: $r$2.
• For $r$3: $r$4.

These extremal graphs exhaust all possibilities, and the corresponding bounds for $r$5 are explicit and sharp.

3. Methodology and Technical Framework

The proofs leverage:

• The structural characterization of $r$6-free graphs via kernel-petal intersections among cliques, ensuring precise vertex overlap across extremal clique families.
• Tensor spectral radius inequalities for nonnegative symmetric tensors (adapting Perron-Frobenius theory to higher orders).
• Weak irreducibility conditions for clique tensors, exploiting $r$7-clique connectivity in the underlying graphs.
• Exchange and deletion arguments that maximize spectral radius without violating forbidden subgraph conditions, notably using the method of vertex neighborhood equalization and supporting calculations via optimization over tensor-induced polynomial forms.

Strong claims, such as uniqueness of extremal graphs and sharp transitions in extremality as $r$8 grows, are rigorously classified.

Implications and Directions

The results generalize the spectral extremal theory for forbidding disjoint cliques from the classical edge-count regime to the non-linear regime of clique tensor spectra. Notably:

• For high $r$9, the spectral radius of clique tensors acts as a proxy for the density and intersection structure of large cliques, providing a fine-grained spectral signature distinguishing extremal graphs even when simple enumeration yields ambiguous results.
• The combinatorial geometry enforced by $\mathcal{A}_r(G)$0-freeness (high clique overlap, large kernel) dovetails directly with spectral concentration in corresponding tensors, establishing a robust combinatorial-spectral correspondence.
• These spectral thresholds can sharpen detection of forbidden substructures in network inference and combinatorial optimization, offering an avenue for tensor-based spectral certification in extremal graph theory.

Theoretically, these results prompt further study in several directions:

• Extensions to higher uniform hypergraphs, leveraging analogous hypergraph adjacency tensors.
• Spectral stability results: quantifying spectral radius drops under small perturbations away from extremal configurations.
• Algorithmic implications: efficient recognition or construction of extremal graphs based on spectral tensor analysis, especially in large $\mathcal{A}_r(G)$1 regimes.

Conclusion

This paper provides a comprehensive spectral characterization of extremal $\mathcal{A}_r(G)$2-free graphs with respect to high-order clique tensor spectral radii. The identification and uniqueness of extremal configurations as certain generalized Turán graphs not only extend existing combinatorial extremal results but also establish new connections between hypergraph spectral theory and forbidden subgraph problems. These findings not only deepen understanding of the interplay between structure and spectra in extremal graph theory but also lay groundwork for further explorations at the interface of spectral methods and combinatorial optimization.