Spectral Turán problem for $t\mathcal{K}_{4}^{-}$-free unbalanced signed graphs

Abstract: Let $tK_4$ denote the family of all graphs consisting of $t$ copies of $K_4$ that are allowed to share vertices and $t\mathcal{K}{4}^{-}$ be the set of all unbalanced signed graphs whose underlying graphs are elements of $tK_4$. In this paper, we characterize the extremal graphs that achieve the maximum index and spectral radius among all $t\mathcal{K}{4}^{-}$-free unbalanced signed graphs with given order.

• The paper establishes that for large n, unbalanced signed graphs free of t copies of K4⁻ achieve a unique maximal spectral index via a specific extremal construction.
• It utilizes switching equivalence, Rayleigh quotient analysis, and equitable partition techniques to derive sharp bounds for both the largest eigenvalue and spectral radius.
• The study extends classical Turán problems to the signed graph setting, offering practical insights for designing networks that avoid undesirable unbalanced substructures.

Spectral Turán Problem for $t\mathcal{K}_4^-$-Free Unbalanced Signed Graphs

Introduction and Problem Context

This paper addresses the spectral Turán problem for unbalanced signed graphs that are free of $t$ copies of the unbalanced signed 4-clique $\mathcal{K}_4^-$—that is, signed graphs whose underlying graphs do not contain $t$ (possibly overlapping) copies of $K_4$ with the induced sign assignment guaranteeing at least one negative cycle in each copy. The study situates itself at the intersection of extremal combinatorics and spectral graph theory for signed graphs, extending classical results on the Turán-type extremal questions to the spectral domain and to the more general signed graph setting.

A signed graph $\Gamma = (G, \sigma)$ consists of a simple undirected graph $G$ together with an edge signature $\sigma : E(G) \to \{+1,-1\}$. The spectral Turán problem seeks, for a given forbidden family $\mathcal{F}$, to determine the maximal index (largest adjacency eigenvalue) or spectral radius among all $\mathcal{F}$-free signed graphs of a given order $t$0. For the family $t$1, which is the set of all unbalanced signed graphs whose underlying graphs are elements of $t$2, the question is: among all $t$3-vertex unbalanced signed graphs that do not contain $t$4 (possibly non-disjoint) $t$5, which graph(s) maximize the spectral index and spectral radius?

The classical unsigned version of the Turán extremal graph problem dates back to Erdős and Turán, and its spectral analog for unsigned graphs was systematically developed by Nikiforov [Nikiforov-LAA-2010]. Various recent works have extended spectral extremal problems to signed graphs, with characterizations for forbidden negative cycles, negative cliques, and more general configurations [Chen-Yuan-AMC-2024, Xiong-Hou-AC-2024]. However, the regime involving multiple forbidden (unbalanced) $t$6 copies introduces new combinatorial and spectral phenomena.

Main Results

The principal achievement of the paper is a characterization of the extremal structure and the precise determination of the maximum index of $t$7-free unbalanced signed graphs of large order $t$8. The main theorem asserts:

• Let $t$9 be an integer and $\mathcal{K}_4^-$0 an integer (the unique solution for which the $\mathcal{K}_4^-$1 contains $\mathcal{K}_4^-$2 $\mathcal{K}_4^-$3 subgraphs).
• For sufficiently large $\mathcal{K}_4^-$4:
  • If $\mathcal{K}_4^-$5, the extremal index is uniquely achieved (up to switching equivalence) by $\mathcal{K}_4^-$6: a signed graph obtained from $\mathcal{K}_4^-$7 by adding a new vertex with exactly $\mathcal{K}_4^-$8 neighbors, where precisely one incident edge is negative.
  • If $\mathcal{K}_4^-$9, the extremal construction becomes the complete signed graph with exactly one negative edge ($t$0).

Explicitly, the maximal index $t$1 among all $t$2-free unbalanced signed graphs is:

$t$3

with equality if and only if $t$4 is switching equivalent to $t$5 or $t$6, respectively.

Additionally, the result extends to the spectral radius $t$7, with the same extremal family under mild further assumptions on $t$8 relative to $t$9.

Methods and Technical Contributions

The proofs synthesize combinatorial, spectral, and switching-equivalence techniques. The argument proceeds via:

• Careful exploitation of switching equivalence: extremal graphs can be assumed to admit a nonnegative principal eigenvector [Sun-Liu-Lan-LAA-2022], allowing the deployment of classical extremal matrix theory (Rayleigh quotient and equitable partitioning).
• Structural reductions leveraging the negative cycle (unbalancedness) to argue that the unique negative edge in the extremal graph can be localized and that any additional negative edge would strictly decrease or not improve the index, contradicting maximality.
• The use of recently established bounds for the spectral index and radius of $K_4$0-free graphs to bound extremal configurations [Xiong-Hou-AC-2024], and the interlacing application of Rayleigh quotient increments and spectral switching arguments.
• The deployment of combinatorial enumeration to relate the parameter $K_4$1 and permitted edge sets, leading to the extremal structure of the underlying graphs.

In the appendix, the authors formalize arguments for the spectral monotonicity in specialized unbalanced signed graphs such as $K_4$2 through equitable partitions and characteristic polynomials of quotient matrices, refining previous extremal results.

Strong Numerical and Contradictory Claims

• The main theorem provides a sharp, explicit construction for the extremal index and characterizes all maximizers up to switching equivalence for sufficiently large $K_4$3.
• The paper demonstrates that for this configuration and parameter range, there is never more than one negative edge in the extremal construction, despite the complex allowable overlap among $K_4$4 subgraphs, which is a nontrivial extension of the unique extremality observed for classical Turán graphs.

Theoretical and Practical Implications

The findings contribute to the theory of extremal signed graphs by:

• Extending the spectral Turán problem from simple graphs to the signed graph context and the more complicated setting of forbidden multiple negative cliques.
• Clarifying the impact of forbidding multiple unbalanced subgraphs on the global behavior of the spectrum and extremal structure; notably, the parameter $K_4$5 and the unique integer root for $K_4$6 play a subtle but decisive role.
• Providing a tool for bounding the largest eigenvalue and spectral properties of signed networks with forbidden frustration or antagonistic substructures, which has implications in the analysis of dynamical processes (e.g., consensus, stability) on signed networks and social balance theory.

On a practical level, the characterization of extremal signed graphs offers a recipe for constructing signed networks with maximized connectivity (as measured by spectral quantities) while avoiding certain types of (possibly undesirable) unbalanced interactions corresponding to negative $K_4$7's.

Future Directions

Potential extensions and open problems include:

• Generalization to $K_4$8 for $K_4$9 or to other classes of forbidden signed subgraphs, using similar methods.
• Determination of sharp thresholds and extremal structures for smaller $\Gamma = (G, \sigma)$0, beyond the asymptotic regime.
• Exploring the impact of multiple negative edges and different balancing patterns within the extremal class, particularly in applications to spectral optimization under signed constraints in real-world networks.
• Investigation of related extremal parameters such as signless Laplacian eigenvalues or singular value spectra for $\Gamma = (G, \sigma)$1-free signed graphs.

Conclusion

The paper provides a detailed resolution of the spectral Turán problem for $\Gamma = (G, \sigma)$2-free unbalanced signed graphs of large order, identifying the unique extremal families maximizing the adjacency spectral index and radius. The authors combine combinatorial and spectral arguments, drawing on switching equivalence and monotonicity, to establish both upper bounds and the precise structure of extremal examples, thus extending fundamental extremal graph theory into the rich domain of signed graphs. The results lay the groundwork for further exploration of spectral extremal questions in the setting of signed and structurally constrained graphs.