Nosal Graphs in Spectral Extremal Theory

• Nosal graphs are triangle-free graphs defined by a spectral radius condition (λ₁(G) ≤ √m) that bridges spectral theory with extremal graph properties.
• They utilize Rayleigh quotient optimization and spectral trace identities to establish tight bounds and stability criteria in extremal graph analysis.
• These structures underpin Turán-type inequalities and supersaturation results, guiding analyses of books, cliques, and cycle counts in graphs.

A Nosal graph is a simple graph at the threshold of spectral triangle-freeness, characterized by the spectral radius of its adjacency matrix not exceeding the square root of its edge count. This notion, introduced by E. Nosal in 1970 as a spectral analog of Mantel’s theorem, forms a cornerstone in spectral extremal graph theory, linking eigenvalue bounds to subgraph containment and providing a concise framework for Turán-type spectral inequalities. The concept and its sharpenings have led to precise characterizations of extremal structures, stability phenomena, and supersaturation theorems for books, cliques, and cycles.

1. Foundational Spectral Bounds

The classical Mantel’s theorem asserts that a graph with $n$ vertices and $m>\frac{n^2}{4}$ edges must contain a triangle. Nosal’s spectral analog replaces the edge threshold with a spectral radius criterion: for a graph $G$ with $m$ edges and largest adjacency eigenvalue $\lambda_1(G)$,

$$\lambda_1(G) > \sqrt{m} \implies \text{G contains a triangle},$$

with equality attainable if and only if $G$ is a complete bipartite graph (possibly with isolates). A Nosal graph is any graph satisfying $\lambda_1(G) \leq \sqrt{m}$; these are precisely the triangle-free graphs not forced by spectral means to contain a triangle (Ning, 2014, Li et al., 19 Nov 2025).

Nikiforov strengthened Nosal’s result: if $\lambda_1(G) \geq \sqrt{m}$, then either $G$ contains a triangle or $G$ is a complete bipartite graph possibly with isolated vertices (Ning, 2014). This sharp spectral threshold connects with the Motzkin–Straus inequality and the Rayleigh quotient arguments, and extensions generalize the triangle case to $K_{r+1}$-free graphs, yielding

$\lambda(G) \leq \sqrt{(1 - 1/r)\,2m},$

with equality on regular complete $r$-partite graphs (Li et al., 19 Nov 2025, Li et al., 2022).

2. Extremal Structures and Equality Characterizations

In the strict sense, Nosal graphs exactly meet the spectral threshold, i.e., $\lambda_1(G) = \sqrt{m}$. The extremal cases are uniquely realized by complete bipartite graphs $K_{a,b}$ with $ab = m$. The Perron vector is constant on each bipartition: $$x_i = \begin{cases} 1/\sqrt{2a} & \text{if } v_i\in \text{part } a, \ 1/\sqrt{2b} & \text{if } v_i\in \text{part } b. \end{cases}$$ Any triangle-free graph maximizing spectral radius, unless it is bipartite, must have structure at or below this threshold (Ning, 2014, Li et al., 2022). For non-bipartite triangle-free graphs, refinements by Lin, Ning, and Wu, Zhai and Shu, and others show the spectral radius obeys even stricter bounds, e.g., $\lambda(G)\leq\sqrt{m-1}$ or $\lambda(G)\leq\beta(m)$, where $\beta(m)$ is the root of an explicit cubic, with equality realized by subdivisions of complete bipartite graphs and certain explicit constructions involving $C_5$ (Li et al., 2022, Li et al., 2023).

3. Supersaturation and Booksize in Nosal Graphs

A major advance is the quantification of substructure supersaturation under the Nosal threshold. If $\lambda(G) > \sqrt{m}$, then not only does $G$ contain a triangle, but it must also contain large “books”—collections of triangles sharing a base edge. Early results established a lower bound

$\mathrm{bk}(G) > \frac{1}{144}\sqrt{m},$

subsequently improved to

$\mathrm{bk}(G) > \frac{1}{9}\sqrt{m},$

for all Nosal graphs with sufficiently many edges (Li et al., 20 Aug 2025, Zhai et al., 15 Jan 2026). The sharpness is illustrated by blow-ups of the triangular prism, demonstrating that the optimal constant is pinned between $1/9$ and $1/3$.

These results rely on delicate eigenvector partitioning, analysis of edge distributions categorized as “good” or “bad,” and iterative edge-deletion or structure/edit distance stability. For non-bipartite $B_{r+1}$-free graphs (with no book of size $r+1$), the extremal configurations include $S_{m,s}^+$ graphs—complete bipartite graphs with one additional edge—and further supersaturation is attainable except for exceptional structures (Zhai et al., 15 Jan 2026).

4. Generalizations to Cliques, Cycles, and Color-Critical Graphs

Nosal’s spectral threshold generalizes to $K_{r+1}$-free graphs, yielding the spectral Turán theorem. The extremal $K_{r+1}$-free graphs maximizing spectral radius are complete $r$-partite Turán graphs. Similar spectral thresholds enforce the occurrence of larger cliques, books, and cycles. Recent extensions also cover color-critical graphs (chromatic number $\chi(F)=r+1$), almost-bipartite graphs, wheels, and theta graphs (Li et al., 19 Nov 2025).

For long cycles, Nikiforov showed that if $\lambda > \sqrt{\lfloor n^2/4\rfloor}$, then all odd cycles of length $3<l\leq n/320$ and (by further refinements) even cycles up to $l\leq \lceil n/28\rceil$ are forced (Ning, 2014).

Further, “book graphs” $B_{r,k}$ generalize the book, counting $K_{r+1}$s sharing an $r$-clique, with their abundance governed by spectral exceedances. Recent results give lower bounds on the number of such shared-clique subgraphs, clarifying conjectures by Nikiforov, Li, Liu, Feng, and others (Li et al., 20 Aug 2025).

5. Spectral Trace Formulae and Triangle Counting Lemmas

The structural connection between eigenvalues and subgraph counts is formalized via spectral trace formulae. For a graph with adjacency matrix $A$,

$$\text{Tr } A^2 = 2m = \sum_{i=1}^n \lambda_i^2,\qquad \text{Tr } A^3 = 6t(G) = \sum_{i=1}^n \lambda_i^3,$$

where $t(G)$ is the triangle count. From these,

$t(G) \geq \frac{1}{3}\lambda_1(\lambda_1^2 - m),$

as shown by Bollobás and Nikiforov, with equality characterizing complete bipartite graphs (Ning et al., 2021, Lin et al., 2019). These trace identities enable spectral-based counting arguments and further stability analysis.

Triangle-free graphs satisfy

$\lambda_1^2 + \lambda_2^2 \leq m,$

settling for $r=2$ the Bollobás–Nikiforov conjecture for clique-free graphs. Equality cases are fully classified via matrix majorization and convexity techniques (Nikiforov, 2021, Lin et al., 2019).

6. Structural Stability, Proof Techniques, and Extremal Analysis

Key methodologies in the study of Nosal graphs include:

• Rayleigh quotient optimization: Utilized to establish spectral upper bounds and reduce to extremal structures with part-constant Perron vectors.
• Spectral Zykov symmetrization: Iteratively replacing non-adjacent vertices by “twins” to maximize symmetry and force extremal configurations (Li et al., 2022).
• Majorization and doubly substochastic matrices: Characterizing possible spectra for triangle-free graphs; equality cases correspond to blow-ups of small paths.
• Interlacing and triangle-counting lemmas: Cauchy interlacing constrains eigenvalue distributions; combined with spectral substructure counts to enforce rigidity.
• Supersaturation and edit-distance stability: Graphs near the spectral threshold must be structurally close to canonical extremal configurations, as shown via edit metrics and Perron vector proximity.

7. Examples, Applications, and Open Directions

Explicit constructions illustrate both the sharpness and rigidity of Nosal thresholds: complete bipartite graphs attain maximal spectral radius without triangles; blow-ups of $C_5$ and subdivision graphs maximize the spectral radius for odd-girth constraints. The spectrum of the adjacency matrix, degree sequences, and eigenvector structures provide combinatorial and algebraic fingerprints of extremal graphs.

Nosal graphs underpin several areas of spectral extremal graph theory, including the development of spectral analogs of Erdős–Rademacher and Mubayi-type theorem, color-critical graph analysis, and stability theory.

Open problems include exact determination of optimal constants in booksize supersaturation, extensions to larger odd-girth and forbidden subgraph families, and spectral proofs of classical extremal results. The general hierarchy of spectral bounds—tracking increasingly restrictive forbidden subgraphs via characteristic polynomials and spectral moments—remains a fertile domain for future investigation (Ning et al., 2021, Zhai et al., 15 Jan 2026).

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Cited papers:

(Ning, 2014, Li et al., 19 Nov 2025, Li et al., 2022, Li et al., 20 Aug 2025, Zhai et al., 15 Jan 2026, Li et al., 2022, Li et al., 2023, Ning et al., 2021, Lin et al., 2019)