A spectral Lovász-Simonovits theorem

A fundamental result in extremal graph theory is attributed to Mantel’s theorem, which states that every graph on $n$ vertices with more than $\lfloor n^2/4 \rfloor$ edges must contain a triangle. Lovász and Simonovits (1975) provided a supersaturation phenomenon by showing that for any $q< n/2$, every graph with $\lfloor n^2/4 \rfloor +q$ edges contains at least $q\lfloor n/2 \rfloor$ triangles. This result resolved a conjecture proposed by Erdős in 1962. In this paper, we establish a spectral counterpart of the result of Lovász and Simonovits. Let $Y_{n,2,q}$ be the graph obtained from the bipartite Turán graph $T_{n,2}$ by embedding a matching with $q$ edges into the partite set of size $\lceil n/2\rceil$. Using the supersaturation-stability method and the spectral techniques, we firstly prove that for $q\le \frac{1}{11}\sqrt{n}$, every graph $G$ on $n$ vertices with spectral radius $λ(G) \ge λ(Y_{n,2,q})$ contains at least $q\lfloor n/2 \rfloor$ triangles. We also show that the bound $q=O(\sqrt{n})$ is tight up to a constant factor, yielding a phenomenon different from that in edge supersaturation. Our result answers a spectral triangle counting problem proposed by Ning and Zhai (2023). Secondly, let $T_{n,2,q}$ be the graph obtained from $T_{n,2}$ by embedding a star with $q$ edges into the partite set of size $\lceil n/2\rceil$. We show further that $T_{n,2,q}$ is the unique extremal graph that contains at most $q\lfloor n/2 \rfloor$ triangles and attains the maximum spectral radius. Thirdly, we present an asymptotic spectral stability result under a specific constraint on the triangle covering number. This result could be viewed as a spectral extension of a recent result proved by Balogh and Clemen (2023), and independently by Liu and Mubayi (2022).

💡 Research Summary

This paper establishes a spectral analogue of the classic Lovász‑Simonovits supersaturation theorem for triangles. The original theorem states that any $n$‑vertex graph with more than $\lfloor n^{2}/4\rfloor+q$ edges contains at least $q\lfloor n/2\rfloor$ triangles, for every $1\le q<n/2$. The authors replace the edge‑count condition by a condition on the spectral radius $\lambda(G)$, the largest eigenvalue of the adjacency matrix.

Two families of extremal graphs are introduced. $Y_{n,2,q}$ is obtained from the bipartite Turán graph $T_{n,2}=K_{\lceil n/2\rceil,\lfloor n/2\rfloor}$ by inserting $q$ pairwise disjoint edges (a matching) into the larger part. $T_{n,2,q}$ is obtained from the same Turán graph by embedding a star $K_{1,q}$ into the larger part. It is easy to verify that $\lambda(Y_{n,2,q})<\lambda(T_{n,2,q})$ for all $q\ge2$.

Theorem 2.2 (spectral Lovász‑Simonovits for matchings) shows that if $1\le q\le \frac{1}{11}\sqrt n$ and $\lambda(G)\ge\lambda(Y_{n,2,q})$, then $G$ contains at least $q\lfloor n/2\rfloor$ triangles. The proof proceeds by (i) deriving tight bounds on $\lambda(Y_{n,2,q})$ via the characteristic polynomial of its adjacency matrix, (ii) using the basic inequality $\lambda(G)\ge 2e(G)/n$ to relate spectral radius to edge count, and (iii) applying a double‑eigenvector technique that forces any graph with such a large spectral radius to be structurally close to $Y_{n,2,q}$. Consequently, the required number of triangles follows from a careful counting argument on the near‑bipartite structure.

Theorem 2.3 strengthens the result: for the same range of $q$, if $\lambda(G)\ge\lambda(T_{n,2,q})$, then $G$ also contains at least $q\lfloor n/2\rfloor$ triangles, and equality holds only when $G\cong T_{n,2,q}$. This establishes $T_{n,2,q}$ as the unique maximizer of the spectral radius among all $n$‑vertex graphs with at most $q\lfloor n/2\rfloor$ triangles. The proof builds on Theorem 2.2 and shows that adding a star rather than a matching yields a strictly larger spectral radius, which in turn forces the extremal structure to be exactly $T_{n,2,q}$.

The authors also investigate the relationship between the spectral radius and the triangle covering number $\tau_{3}(G)$ (the size of a smallest vertex set intersecting every triangle). Theorem 2.5 proves that for any $n\ge 28s^{2}$, if $\lambda(G)\ge\lambda(T_{n,2})$ and $\tau_{3}(G)\ge s$, then $G$ contains at least $\frac12sn-5s^{2}$ triangles. This is a spectral extension of recent stability results by Balogh‑Clemen and Liu‑Mubayi, which gave edge‑based lower bounds under the same covering‑number condition. The proof combines the spectral lower bound $\lambda(G)\ge 2e(G)/n$ with the supersaturation bound of Theorem 2.2, yielding a linear‑in‑$n$ lower bound on the number of triangles when the covering number is fixed.

An important aspect of the work is the identification of the optimal range $q=O(\sqrt n)$. The authors construct examples showing that when $q$ exceeds a constant multiple of $\sqrt n$, the spectral radii of $Y_{n,2,q}$ and $T_{n,2,q}$ become essentially indistinguishable, and the triangle lower bound $q\lfloor n/2\rfloor$ can no longer be guaranteed. This contrasts sharply with the edge‑based supersaturation, where $q$ may be as large as $n/2$.

Methodologically, the paper blends classical extremal graph theory with spectral techniques. The key tools are: (1) precise eigenvalue calculations for the two families of extremal graphs, (2) the Rayleigh quotient and the inequality $\lambda(G)\ge 2e(G)/n$, (3) a double‑eigenvector argument that forces a near‑bipartite structure for graphs with large spectral radius, and (4) careful counting of triangles in graphs that are close to the extremal constructions.

Overall, the paper provides a comprehensive spectral counterpart to the Lovász‑Simonovits theorem, identifies the exact extremal graphs for the spectral problem, proves uniqueness, and extends the analysis to stability under triangle covering constraints. It opens a new direction for spectral supersaturation problems involving other color‑critical subgraphs, cliques, cycles, and beyond.