Brualdi–Solheid Problem in Spectral Graphs

• The Brualdi–Solheid Problem is defined as finding the maximum or minimum spectral radius in graph families constrained by parameters like forbidden subgraphs and degree limits.
• It generalizes Turán-type extremal graph theory by linking edge count optimization with eigenvalue methods, often identifying threshold or Turán graphs as extremals.
• Key methodologies include the use of Rayleigh quotient arguments, eigenvalue interlacing, and structural reductions that transform graph configurations to enhance spectral properties.

The Brualdi–Solheid Problem refers to a class of extremal spectral graph theory questions concerning the maximum or minimum possible spectral radius of graphs within a specified family, typically subject to discrete structural constraints such as order, size, degree, forbidden subgraphs, or other invariants. This family of problems generalizes classical extremal graph theory (Turán-type) to the spectral field, and has become central in connecting combinatorial and linear algebraic aspects of graph structure.

1. Problem Formulation and Fundamental Concepts

For a family of graphs $\mathscr{H}$ defined by a property (e.g., being $F$-free, fixed order/size, degree constraints), the Brualdi–Solheid Problem asks:

$\max\{\rho(G): G \in \mathscr{H}\}$

where $\rho(G)$ denotes the spectral radius (i.e., the largest eigenvalue) of the adjacency matrix of $G$; in some variants, one considers other eigenvalue functions, such as the signless Laplacian or the general $A_\alpha$-index.

Key definitions include:

• $F$-free graph: $G$ is $F$-free if it contains no subgraph isomorphic to $F$.
• Extremal graph: A graph achieving the maximum (or minimum) spectral radius in $\mathscr{H}$.
• Classical Turán function (edge extremal): $$\mathrm{ex}(n, F) = \max\{e(G): |G|=n,\, G \text{ is } F \text{-free}\}$$
• Brualdi–Solheid Function ("spectral Turán"): $$\mathrm{spex}(n, F) = \max\{\rho(G): |G|=n,\, G \text{ is } F \text{-free}\}$$

The Brualdi–Solheid problem also appears in variants maximizing $\rho(G)$ subject to fixed order and size, degree constraints, connectivity requirements, or forbidden minors (Ren et al., 2023, Zhang et al., 10 Nov 2025, Wang et al., 2024, Csikvári et al., 2024).

2. Classical Cases and Methodological Reductions

The original instance was posed for the class $\mathcal{H}_{n,m}$: all connected simple graphs with $n$ vertices and $m$ edges. In this setting, extremal graphs maximizing $\rho(G)$ are always threshold graphs, a class defined by a recursive construction or an explicit vertex-ordering with associated generating binary sequence (Thm., (Csikvári et al., 2024)).

For forbidden subgraphs, the paradigm is parallel to classical Turán theory: maximizing $\rho(G)$ over all $n$-vertex $F$-free graphs. Extensively, solutions show that, in many cases, the extremal graphs are the same as in the edge extremal (Turán) setting; i.e., for large $n$ the same $r$-partite Turán graphs or their explicit joins (Fang et al., 2023, Ren et al., 2023, Zhang et al., 2021, Zhao et al., 2020).

For generalizations such as the $A_\alpha$-matrix, the same classification for $A(G)$-extremals persists for $A_\alpha(G)$ across small- or moderate-size regimes, with threshold graphs again playing the key role (Zhang et al., 10 Nov 2025).

A common methodological insight is the reduction to monotone degree sequences and structural transformations (edge/vertex swaps) that strictly increase $\rho(G)$ unless the underlying structure is stabilized—this forces extremality conditions and reduces complexity, especially in connected/threshold cases (Csikvári et al., 2024, Zhang et al., 10 Nov 2025).

3. Forbidden Subgraph Formulations: Spectral Turán Problems

In the forbidden subgraph context, the Brualdi–Solheid problem asks for the spectral counterpart to classical Turán-type results:

• For $F=C_\ell^2$, the square of the $\ell$-cycle with $\ell\ge6$ and $\ell\not\equiv0\pmod3$, the extremal graph for both maximal edge count and maximal spectral radius is $T_{n,3}$ (balanced complete 3-partite graph) for $\ell\equiv1\pmod3$, and the join $K_1+T_{n-1,3}$ for $\ell\equiv2\pmod3$ (Fang et al., 2023).
• For even cycles $C_{2k+2}$ and trees on $2k+3$ vertices in bipartite graphs, the unique extremal graphs are the complete bipartite $K_{k,n-k}$ (Ren et al., 2023).
• For consecutive odd cycles $C_{2\ell+1}$ up to a fixed length in non-bipartite graphs, the extremal structure is $K_{a, b}\bullet K_{3}$, the complete bipartite $K_{a,b}$ with a triangle "grafted" via vertex identification (Zhang et al., 2021).

The spectral extremal problem for wheel-free graphs—graphs not containing any $W_k$ for $k \geq 4$—has also been completely resolved, characterizing the extremal family $H_n$ depending on $n \pmod{4}$ and, for the signless Laplacian, a distinct join structure $K_2 \nabla (n-2)K_1$ (Zhao et al., 2020).

A recurring theme is that, for most standard forbidden subgraphs (cycles, trees, wheels, and certain bipartite structures), the extremal graphs for the spectral radius mirror those for extremal edge count, with only exceptional configurations or parity restrictions yielding distinct extremals.

4. Spectral Extremal Results for Trees and Generalized Parameters

For the family of all graphs with given matching number or vertex/edge-connectivity properties, the Brualdi–Solheid problem admits both minimization and maximization forms:

• Graphs with given matching number $\beta$: The minimizer for the spectral radius over $\mathcal{G}_{n,\beta}$ is always a tree, structurally classified via the novel quasi-adjacency relation that encodes clique block-decomposition among control sets (Theorem 2.6 etc., (Liu et al., 26 Jan 2026)). Extremal trees are constructed by balancing the sizes of attached stars and central paths; explicit formulas are provided for $\beta=2,3,4$.
• Graphs with prescribed $h$-extra $r$-component connectivity: For $\mathcal{G}_{n,\delta}^{\kappa_r^h}$, extremal graphs maximizing $\rho(G)$ are unique, characterized by a join of cliques with minimal spread of missing edges to achieve the required degree and cut conditions (Wang et al., 2024).
• Connected threshold graphs with fixed order and size: Brualdi–Solheid extremals are always threshold graphs, and the paper provides tight spectral upper and lower bounds in terms of parameters of the binary generating sequence and the structure of "lazy walks" (Csikvári et al., 2024).

5. Core Techniques and Major Theoretical Insights

• Eigenvalue and Rayleigh Quotient Argumentation: Critical for establishing extremality under spectral optimization, especially when coupled with monotonicity properties under local modifications (edge addition/move increases $\rho(G)$ unless structural balance is achieved) (Zhang et al., 10 Nov 2025, Wang et al., 2024).
• Structural Reductions to Threshold Graphs: In fixed $(n,m)$ connected settings, extremality is only possible for threshold graphs, which admit a recursive construction and simple degree ordering (Csikvári et al., 2024, Zhang et al., 10 Nov 2025).
• Spectral Stability and Perturbation Lemmas: For forbidden subgraph problems, small deviations from extremality in the spectral radius imply $o(n^2)$-edge differences from the conjectured extremal template, enforceable via Perron vector analysis and edge-swapping (Fang et al., 2023).
• Combinatorial–Algebraic Interplay: Explicit use of eigenvalue interlacing, vertex partitioning by Perron vector entries, and local combinatorial arguments (path/cycle embeddings, component decompositions) to translate global spectral constraints into strong structural characterizations (Ren et al., 2023, Zhao et al., 2020, Zhang et al., 2021).
• Characteristic Polynomial Engineering: In the matching-minimal case, recurrence relations and block-gluing formulas for characteristic polynomials yield precise control over the possible spectral radii for candidate trees (Liu et al., 26 Jan 2026).

6. Open Problems and Directions

• The Brualdi–Solheid maximal spectral radius problem for all $n, m$ and for certain families of forbidden subgraphs (especially for intermediate or large $m$) remains unresolved and may require refined local-global structural techniques beyond threshold graph reductions (Zhang et al., 10 Nov 2025, Csikvári et al., 2024).
• For $A_\alpha$-spectral radius with general $\alpha \in [0,1)$, characterizations are complete only for substantial ranges of $m$; structure for near-complete or dense graphs is less well-understood.
• Extension to spectral extremal problems for the second largest eigenvalue, spectral gaps, and non-adjacency matrices (e.g., normalized Laplacian) is largely undeveloped (Zhang et al., 10 Nov 2025).
• For minimization in fixed-invariant classes (such as matching number), rich combinatorial phenomena linking quasi-adjacency, domination, and balancing constraints suggest broader block-graph structural classifications (Liu et al., 26 Jan 2026).

7. Position and Impact in Extremal Graph Theory

The Brualdi–Solheid Problem has unified classical combinatorial extremal theory with algebraic (spectral) methods, revealing strong connections between global spectral parameters and fine graph structure. For a wide array of forbidden subgraphs and invariants, extremal graphs in the spectral setting are determined and often coincide with their combinatorial (edge-count extremal) counterparts, although spectral analysis sometimes uncovers exceptional extremals inaccessible to purely combinatorial arguments (Fang et al., 2023, Ren et al., 2023, Zhao et al., 2020, Zhang et al., 10 Nov 2025, Csikvári et al., 2024, Wang et al., 2024, Liu et al., 26 Jan 2026). This suggests an underlying universality and hints at further algebraic symmetries in graph structure yet to be fully described.