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Yongtao Li

Publications and source records attributed to Yongtao Li.

At least 19 recordsLinked to original sources
arXiv2026

Edge-spectral Turán theorems for color-critical graphs with applications

A classical result of Nosal asserts that every $m$-edge graph with spectral radius $λ(G)> \sqrt{m}$ contains a triangle. A celebrated extension of Nikiforov [2002, CPC] states that if $G$ is an $m$-edge graph with $λ(G)> \sqrt{(1- {1}/{r})2m}$, then $G$ contains a clique $K_{r+1}$. This result implies the Turán theorem and Wilf theorem, and offers a new perspective on the existence of substructures. The edge-spectral conditions are versatile for enforcing substructures, as they can be applied to any sparse graph regardless of its edge density. In this paper, we prove that for any color-critical graph $F$ with chromatic number $χ(F)=r+1\ge 4$, if $m$ is sufficiently large and $G$ is an $F$-free graph with $m$ edges, then $λ(G)\le \sqrt{(1- {1}/{r})2m}$, with equality if and only if $G$ is a regular complete $r$-partite graph. This settles an open problem proposed by Yu and Li [2025, arXiv] and also gives spectral bounds for graphs forbidding books and wheels. Secondly, we establish an asymptotic formula and structural characterization when we forbid an almost-bipartite graph $F$, where $F$ is called almost-bipartite if it can be made bipartite by removing at most one edge. As applications, we determine the unique $m$-edge spectral extremal graph for every integer $m$ when avoiding certain substructures, including complete bipartite graphs plus an edge, cycles plus an edge, and theta graphs, etc. Our results resolve an open problem proposed by Li, Zhao and Zou [2025, JGT], as well as two conjectures posed by Liu and Li [2025, LAA]. The arguments in our proofs are based on the edge-spectral stability method recently established by the authors. In addition, we develop some new spectral techniques, including the stability result for the Perron--Frobenius eigenvector.

math.CO↗
arXiv2026

Chiral classical and quantum acoustics with hole-spin qubits

Gate-defined hole spins combine strong spin-orbit coupling with exceptional strain sensitivity, making surface acoustic waves a natural route to remote, phase-coherent control. We show that counterpropagating surface acoustic waves can differ in coupling strength and in whether they predominantly drive spin rotations or modulate the qubit frequency. We call the transverse transition-strength imbalance spin-acoustic chirality and show that it is tunable through gate-controlled reshaping of the dot confinement. Combining a multiband Luttinger-Kohn Hamiltonian with the Bir-Pikus description of strain and 3D piezoelectric finite-element simulations, we obtain the acoustic g-matrix modulation governing coherent SAW driving. Quantizing the same strain-mediated interaction yields the corresponding single-phonon coupling, and a generalized anisotropic Rabi interaction with co-rotating, counter-rotating, and longitudinal components, linking classical chirality to quantum anisotropy and directional one-phonon emission. Magnetic field orientation and dot shape tune the acoustic chirality and operator selectivity. We illustrate this framework through bichromatic coherence protection and phonon-mediated heralded Bell-state initialization.

cond-mat.mes-hall↗
arXiv2026

An entropy bridge from weighted to spectral Turán theorems

We establish an entropy bridge, and then use it to prove that for any color-critical graph $F$ with chromatic number $χ(F)=r+1\ge 3$, there exists a constant $λ_0=λ_0(F)$ such that if $G$ is an $F$-free graph with $λ(G)\ge λ_0$, then for every $\ell\ge 1$ with $(r,\ell )\neq (2,2)$, \[ λ^\ell (G) \le \Big(1-\frac1r\Big)w_\ell(G), \] with equality if and only if $G$ is a regular complete $r$-partite graph; in the case $r=2$ with $\ell$ even, equality holds for every complete bipartite graph. The pair $(r,\ell )=(2,2)$ must be excluded, since the bound fails for several forbidden graphs, e.g., $C_{2t+1}$ with $t\ge 2$. Moreover, walks counts may also be replaced by the homomorphism counts of unbalanced trees. As further applications, we extend a spectral supersaturation result of Bollobás and Nikiforov [J. Combin. Theory Ser. B (2007)], and we also extend the entropic Turán theorem of Chao and Yu [J. London Math. Soc. (2026)] from $K_{r+1}$-free graphs to $F$-free graphs with $F$ color-critical. We provide a framework by passing through weighted Turá theorems of independent interest. If $G$ is $F$-free and $\mathbf{p}$ is a probability vector on $V(G)$ with $\lVert \mathbf{p}\rVert_\infty$ sufficiently small, then \[ 2\sum_{uv\in E(G)}p_up_v \le 1-\frac1r + o(1), \] and the error term $o(1)$ can be removed if and only if $F$ is color-critical. This is a Motzkin-Straus-type inequality in which the clique number of $G$ is replaced by $χ(F)-1$. The proof of this weighted result combines a blow-up argument, the graph removal lemma, the Erdős-Simonovits stability theorem, a probabilistic sampling argument, and an exact estimate near a complete $r$-partite graph. The bridge linking spectral inequalities to weighted inequalities is based on the entropy method for the Markov chain attached to the Perron vector.

math.CO↗
arXiv2026

The product measures of cross $t$-intersecting families

We investigate the product measures of intersection problems in extremal combinatorics. Invoking a recent result of He--Li--Wu--Zhang, we prove that for any $ n \geq t \geq 3$ and $ p_1, p_2 \in (0, \frac{1}{t+1})$, if $ \mathcal{F}_1, \mathcal{F}_2 \subseteq 2^{[n]}$ are cross $ t$-intersecting families, then $μ_{p_1}(\mathcal{F}_1)μ_{p_2}(\mathcal{F}_2)\le (p_1p_2)^t$. Secondly, we study the intersection problems for integer sequences by proving that if $\mathcal{H}_1, \mathcal{H}_2 \subseteq [m]^{n}$ are cross $t$-intersecting with $ m > t+1$, then $|\mathcal{H}_1|| \mathcal{H}_2|\leq (m^{n-t})^2$. These results confirm two classical conjectures of Tokushige. As an application, we strengthen a recent theorem of Frankl--Kupavskii, generalizing the well-known IU-Theorem. Finally, we show that if $ p \geq \frac{1}{2}$ and $ \mathcal{F}_1, \mathcal{F}_2 \subseteq 2^{[n]}$ are cross $t$-intersecting families, then $\min \left\{μ_{p}(\mathcal{F}_1),μ_{p}(\mathcal{F}_2)\right\} \leq μ_{p}(\mathcal{K}(n,t))$, where $\mathcal{K}(n,t)$ denotes the Katona family. This recovers an old result of Ahlswede--Katona.

math.CO↗
arXiv2026

Non-uniform pairwise cross $t$-intersecting families

Let $ n\geq t\geq 1$ and $ \mathcal{A}_1, \mathcal{A}_2, \ldots, \mathcal{A}_m \subseteq 2^{[n]}$ be non-empty families. We say that they are pairwise cross $t$-intersecting if $|A_i\cap A_j|\geq t$ holds for any $A_i\in \mathcal{A}_i$ and $A_j\in \mathcal{A}_j$ with $i\neq j$. In the case where $m=2$ and $\mathcal{A}_1=\mathcal{A}_2$, determining the maximum size $M(n,t)$ of a non-uniform $t$-intersecting family of sets over $[n]$ was solved by Katona (1964), and enhanced by Frankl (2017), and recently by Li and Wu (2024). In this paper, we establish the following upper bound: if $ \mathcal{A}_1, \mathcal{A}_2, \ldots, \mathcal{A}_m \subseteq 2^{[n]}$ are non-empty pairwise cross $t$-intersecting families, then $$ \sum_{i=1}^m |\mathcal{A}_i| \leq \max \left\{ \sum_{k=t} ^{n}\binom{n}{k} + m - 1, \, m M(n, t) \right\}. $$ Furthermore, we provide a complete characterization of the extremal families that achieve the bound. Our result not only generalizes an old result of Katona (1964) for a single family, but also extends a theorem of Frankl and Wong (2021) for two families. Moreover, our result could be viewed as a non-uniform version of a recent theorem of Li and Zhang (2025). The key in our proof is to utilize the generating set method and the pushing-pulling method together.

math.CO↗
arXiv2026

Supersaturation in Nosal graphs: Triangles and books

In this paper, we use the spectral surplus $λ(G) - \sqrt{m}$ to measure how far $G$ lies above the Nosal threshold, and prove the following edge-spectral supersaturation results for triangles and books. (a) Every graph $G$ with $m\ge 3$ edges and $λ(G) \ge 1 + \sqrt{m-2}$ contains at least $m-2$ triangles, with equality if and only if $G = K_3 \vee \tfrac{m-3}{3} K_1$. This can be viewed as the third-layer supersaturation in the jump phenomenon, after the first layer $t(G) \ge \lfloor \tfrac{1}{2}(\sqrt{m}-1) \rfloor$ proved by Ning and Zhai, and the second layer $t(G) \ge \tfrac{m-1}{2}$ by Zhang and Zhai. (b) Every $m$-edge graph $G$ satisfies $t(G) \ge m\bigl(λ- \sqrt{m}\,\bigr)$, with equality if and only if $G$ is complete bipartite. Consequently, $λ(G) \ge \sqrt{m} + q$ forces $t(G) > q m$ for every real $q > 0$. This is an edge-spectral counterpart of the Lovász--Simonovits theorem, and it improves the Bollobás--Nikiforov bound $t(G) \ge \tfrac13 λ(λ^2 - m)$ in the range $\sqrt m \le λ(G) \le 1.3\sqrt m $. (c) Every $m$-edge Nosal graph $G$ contains a book of size greater than $\tfrac14 \sqrt{m}$. This improves two recent results on the booksize constant: $\tfrac{1}{24}$ proved by Li, Liu and Zhang, and $\tfrac19$ by Zhai, Li and Lou. This narrows the gap toward the conjectured optimal constant $\tfrac13$. (d) Every $m$-edge Nosal graph $G$ contains at least $\bigl(\tfrac{1}{8} - o(1)\bigr) m$ copies of the kite $C_4^+=B_2$, and the constant $\tfrac18$ is best possible. This determines the sharp asymptotic constant for counting $C_4^+$ and strengthens the $Ω(m)$ bound of Li, Liu and Zhang.

math.CO↗
arXiv2026

An edge-spectral supersaturation of Mubayi's theorem for color-critical graphs

We study the supersaturation problem in its edge-spectral form. Let $λ(G)$ be the adjacency spectral radius of $G$. Nikiforov proved that every $K_{r+1}$-free graph $G$ with $m$ edges satisfies $λ(G)\le \sqrt{(1\!-\!1/r )2m}$. Recently, Li, Liu and Zhang proved the same bound for every $F$-free graph $G$, where $F$ is any color-critical graph with $χ(F)=r+1\ge4$, with equality only for regular complete $r$-partite graphs. It is then natural to ask how many copies of $F$ are forced once $λ(G)$ exceeds this threshold. Fang, Lin and Zhai answered this at the threshold itself, and conjectured that for any fixed $C>0$, the condition $λ(G)\ge \sqrt{(1\!-\!{1}/{r})2m} +C$ forces $Ω\!\left(m^{(f-1)/2}\right)$ copies. In this paper, we answer this question with the best possible constant. Building on the proof framework of Fang, Lin and Zhai, we prove that for every color-critical graph $F$ with $χ(F)=r+1\ge4$, there exists $δ_F>0$ such that if $m$ is sufficiently large, $0<q\leδ_F\sqrt m$, and $G$ is an $m$-edge graph with $λ^2(G)\ge 2\left(1-\tfrac1r\right)m+q$, then \[ N_F(G)\ge\bigl(B_F-o(1)\bigr)\,q\, m^{{(f-2)}/{2}}, \quad \text{where}~~ B_F:=\tfrac{α_F}{4} (\tfrac{2r}{r-1} )^{{f}/{2}}, \] and the constant $B_F$ is best possible. Our result can be viewed as an edge-spectral counterpart of Mubayi's theorem, since it converts the spectral surplus $q$ into a linear number of copies of $F$, and it solves the conjecture of Fang, Lin and Zhai in a stronger form.

math.CO↗
arXiv2026

Spectral Sidorenko inequalities and edge-spectral supersaturation

We develop a spectral approach to Sidorenko-type inequalities and apply it to establish sharp edge-spectral supersaturation results. Let $H$ be a bipartite graph with $v$ vertices and $e$ edges, where $v\le e$, and write $M(G)=2e(G)$. We prove that Sidorenko's conjecture is equivalent to a spectral strengthening: $$ \hom(H,G)\ge M(G)^e |V(G)|^{v-2e} \quad \text{ if and only if }\quad \hom(H,G)\ge λ(G)^{2e-v}M(G)^{v-e}.$$ We also introduce an operator-norm certificate which, via the Riesz--Thorin interpolation, gives direct proofs of the spectral Sidorenko inequality in several cases. The converse direction in the equivalence theorem is proved by a tensor-power spectral regularization lemma. Our main result provides a unified framework to prove sharp asymptotic edge-spectral supersaturation results for degenerate bipartite graphs with the Sidorenko property, including complete bipartite graphs and even cycles. Let $S_{t-1,m}$ be the split graph with $m$ edges obtained by joining a clique $K_{t-1}$ to an independent set. For every $m$-edge graph $G$ with $λ(G)>λ(S_{t-1,m})$, $$\texttt{#} K_{t,t}(G) \ge \Big(\frac{2^{-(t-1)^2}}{(t!)^2}-o(1)\Big)m^t \quad \text{and}\quad \texttt{#}C_{2t}(G) \ge \Big(\frac{(t-1)!}{2t^t}-o(1)\Big)m^t.$$ Both constants are best possible: the first is attained asymptotically by random graphs, while the second is attained by split graphs. The supersaturation proofs combine spectral Sidorenko inequalities with a heavy-edge pruning process, a Perron-vector localized/delocalized dichotomy, and incidence-matrix inequalities.

math.CO↗
arXiv2026

On a conjecture of distance spectral extremal problems

Brualdi and Hoffman proposed a well-known problem of determining the graph with maximum adjacency spectral radius among all graphs with given size $m$. Early work by Friedland and Stanley addressed some specific cases. This problem was later completely solved by Rowlinson and recently revisited by Cheng and Weng. Pioneering work on the distance matrix was carried out by Graham and Pollak, as well as by Graham and Lovász. The distance spectral radius $ρ(G)$ of a connected graph $G$ is the largest eigenvalue of its distance matrix. In this paper, we completely solve the problem of characterizing the connected graph with minimum distance spectral radius among all graphs with size $m$. Let $\mathcal{G}(m)$ be the class of connected graphs with $m$ edges. For every $m \ge 3$, let $n$ be the unique integer satisfying $\binom{n-1}{2} < m \le \binom{n}{2}$, and we write $m = \binom{n-1}{2} + s$ with $1 \le s \le n-1$. Recently, Lin and Zhou [Adv. in Appl. Math. 173 (2026)] investigated the graph in $\mathcal{G}(m)$ that minimizes $ρ(G)$ in the range $s \ge \max\{ \frac{n-6}{2}, 1\}$. However, the problem is much more difficult in the remaining range $1 \le s \le \frac{n-7}{2}$, and they conjectured that the unique minimizer is $\overline{P_{n,s+1}}$, the complement of a balanced disjoint union of paths. Using novel matrix analysis, we solve this conjecture in the affirmative. Moreover, we provide a new unified proof for the entire range $1 \le s \le n-1$. The key ingredients in our proof argument include an innovative comparison principle for the distance spectral radius, an increment analysis of $Φ$-functions on paths and cycles, an argument for balancing path lengths, and a walk enumeration technique via the Neumann series.

math.CO↗
arXiv2026

An exponentially small gap of the Perron vector on independent sets

A classical result of Cioabă states that if $G$ is a connected graph with the unit Perron vector $\mathbf{x}$, then any independent set $S$ of $G$ satisfies $\sum_{v\in S} x_v^2 \le \frac{1}{2}$, with equality if and only if $G$ is a bipartite graph and $S$ is one of the partite sets. Let $χ(G)= k $ be the chromatic number of $G$. A well-known conjecture of Gregory asserts that any independent set $S$ of $G$ satisfies $\frac{1}{2} - \sum_{v\in S}x_v^2 = Ω((k/n)^{1/2})$. Recently, Liu and Ning [J. Combin. Theory Ser. B 176 (2026)] disproved Gregory's conjecture by constructing a graph $G$ and an independent set $S$ such that $\frac{1}{2}- \sum_{v\in S}x_v^2 = O(k^5/n^3)$. Furthermore, they conjectured that this bound is tight up to a constant factor. In this paper, we first show that any cycle $C_n$ with odd integer $n\ge 7$ provides a simple counterexample to Gregory's conjecture. Second, we establish that for any independent set $S$, we have $\frac{1}{2} - \sum_{v\in S}x_v^2 = \frac{q}{4λ-2q}$, where $λ$ is the spectral radius of $G$, and $q$ is the Rayleigh quotient of $\mathbf{x}$ restricted to $\bar{S} :=V(G)\setminus S$. Third, we construct a graph with arbitrarily large chromatic number and find an independent set $S$ such that $\sum_{v\in S}x_v^2$ can be arbitrarily close to $\frac{1}{2}$, with an exponentially small gap. Our construction shows that there is no universal lower bound of the form $Ω(k^α/n^β)$ for any $α, β>0$. This settles both Gregory's original conjecture and the modified conjecture of Liu and Ning in the negative. Finally, we show the tightness of our construction and provide some local weighted lower bounds.

math.CO↗
arXiv2026

Two results on set families: sturdiness and intersection

This paper resolves two open problems in extremal set theory. For a family $\mathcal{F} \subseteq 2^{[n]}$ and $i, j\in [n]$, we denote $\mathcal{F} (i,\bar{j})=\{F\backslash\{i\}: F\in \mathcal{F}, F\cap\{i,j\}=\{i\}\}$. The sturdiness $β(\mathcal{F})$ is defined as the minimum $|\mathcal{F} (i,\bar{j})|$ over all $i\neq j$. A family $\mathcal{F}$ is called an IU-family if it satisfies the intersection constraint: $F\cap F'\neq \emptyset $ for all $F,F'\in \mathcal{F}$, as well as the union constraint: $F\cup F' \neq [n]$ for all $F,F'\in \mathcal{F}$. The well-known IU-Theorem states that every IU-family $\mathcal{F}\subseteq 2^{[n]}$ has size at most $ 2^{n-2}$. In this paper, we prove that if $\mathcal{F}\subseteq 2^{[n]}$ is an IU-family, then $β(\mathcal{F})\le 2^{n-4}$. This confirms a recent conjecture proposed by Frankl and Wang. As the second result, we establish a tight upper bound on the sum of sizes of cross $t$-intersecting separated families. Our result not only extends a previous theorem of Frankl, Liu, Wang and Yang on separated families, but also provides explicit counterexamples to an open problem proposed by them, thereby settling their problem in the negative.

math.CO↗
arXiv2026

Some Turán-type results for the signless Laplacian spectral radius

Half a century ago, Bollobás and Erdős [Bull. London Math. Soc. 5 (1973)] proved that every $n$-vertex graph $G$ with $e(G)\ge (1- \frac{1}{k} + \varepsilon )\frac{n^2}{2}$ edges contains a blowup $K_{k+1}[t]$ with $t=Ω_{k,\varepsilon}(\log n)$. A well-known theorem of Nikiforov [Combin. Probab. Comput. 18 (3) (2009)] asserts that if $G$ is an $n$-vertex graph with adjacency spectral radius $λ(G)\ge (1- \frac{1}{k} + \varepsilon)n$, then $G$ contains a blowup $K_{k+1}[t]$ with $t=Ω_{k,\varepsilon}(\log n)$. This gives a spectral version of the Bollobás--Erdős theorem. In this paper, we systematically explore variants of Nikiforov's result in terms of the signless Laplacian spectral radius, extending the supersaturation, blowup of cliques and the stability results.

math.CO↗
arXiv2026

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).

math.CO↗
arXiv2026

Subspace variations of the weighted skew Bollobás theorem

Let $V$ be a finite-dimensional real vector space. A collection $\mathcal{P} = \{(A_i,B_i)\}_{i=1}^m$ of pairs of subspaces of $V$ is called a skew Bollobás system if $\dim(A_i\cap B_i)=0$ for each $i\in [m]$ and $\dim(A_i\cap B_j)>0$ for all $1\leq i<j \leq m$. Assume that $V = V^{(1)}\oplus \cdots \oplus V^{(r)}$ and $\mathcal{P}= \{(A_i,B_i)\}_{i=1}^m$ is a skew Bollobás system of subspaces of $V$ satisfying $ A_i = \bigoplus_{k=1}^r (A_i \cap V^{(k)})$ and $ B_i = \bigoplus_{k=1}^r (B_i \cap V^{(k)})$ for each $i\in [m]$. Denote $a_{i,k} = \dim(A_i \cap V^{(k)})$ and $b_{i,k} = \dim(B_i \cap V^{(k)})$. Suppose that $a_{1,k} \le \cdots \le a_{m,k}$ and $b_{1,k} \ge \cdots \ge b_{m,k}$ for each $k\in [r]$. Using the exterior algebraic method developed by Lovász and Scott--Wilmer, we prove that $$ \sum_{i=1}^{m} \frac{1}{\prod_{k=1}^{r} \binom{a_{i,k}+b_{i,k}}{a_{i,k}}} \le 1 . $$ This generalizes the results of Alon (JCTA, 1985) and Scott--Wilmer (JLMS, 2021) to multipart weighted setting. Secondly, we solve a conjecture of Hegedüs (AJC, 2015) concerning projective subspaces, showing that any skew Bollobás system of projective subspaces in an $n$-dimensional projective space contains at most $2^{n+1} - 2$ pairs. Thirdly, we prove that if $\mathcal{P}= \{(A_i,B_i)\}_{i=1}^m$ is a skew Bollobás system of subspaces of $V$ with $a_i=\dim (A_i)$ and $b_i=\dim (B_i)$, then $$ \sum_{i=1}^m \frac{1}{(a_i+ b_i+1)\binom{a_i+b_i}{a_i}} \le 1. $$ This gives an extension to the subspace setting of the results of Hegedüs--Frankl (EUJC, 2024) and Yue (DM, 2026). Finally, we extend the above inequality to systems of $d$-tuples of subspaces, giving a unified bound that implies the corresponding results for $d$-tuples of subsets.

math.CO↗
arXiv2026

More on spectral supersaturation for the bowtie

A central topic in extremal graph theory is the supersaturation problem, which studies the minimum number of copies of a fixed substructure that must appear in any graph with more edges than the corresponding Turán number. Significant works due to Erdős, Rademacher, Lovász and Simonovits investigated the supersaturation problem for the triangle. Moreover, Kang, Makai and Pikhurko studied the case for the bowtie, which consists of two triangles sharing a vertex. Building upon the pivotal results established by Bollobás, Nikiforov, Ning and Zhai on counting triangles via the spectral radius, we study in this paper the spectral supersaturation problem for the bowtie. Let $λ(G)$ be the spectral radius of a graph $G$, and let $K_{\lceil \frac{n}{2}\rceil, \lfloor \frac{n}{2}\rfloor}^q$ be the graph obtained from Turán graph $T_{n,2}$ by adding $q$ pairwise disjoint edges to the partite set of size $\lceil \frac{n}{2}\rceil$. Firstly, we prove that there exists an absolute constant $δ>0$ such that if $n$ is sufficiently large, $2\le q \le δ\sqrt{n}$, and $G$ is an $n$-vertex graph with $λ(G)\ge λ(K_{\lceil \frac{n}{2}\rceil, \lfloor \frac{n}{2}\rfloor}^q)$, then $G$ contains at least ${q\choose 2}\lfloor \frac{n}{2}\rfloor$ bowties, and $K_{\lceil \frac{n}{2}\rceil, \lfloor \frac{n}{2}\rfloor}^q$ is the unique spectral extremal graph. This solves an open problem proposed by Li, Feng and Peng. Secondly, we show that a graph $G$ whose spectral radius exceeds that of the spectral extremal graph for the bowtie must contain at least $\lfloor \frac{n-1}{2}\rfloor$ bowties. This sharp bound reveals a distinct phenomenon from the edge-supersaturation case, which guarantees at least $\lfloor \frac{n}{2}\rfloor$ bowties.

math.CO↗
arXiv2025

Spectral supersaturation for color-critical graphs

A graph is color-critical if it contains an edge whose deletion reduces its chromatic number. This class of graphs, including cliques and odd cycles, plays a central role in extremal graph theory. In this paper, following an influential line of research initiated by Bollobás-Nikiforov, we study the spectral supersaturation problem for color-critical graphs. Let $T_{n,r}$ be the $r$-partite Turán graph, let $\mathcal{T}_{n,r,q}$ denote the family of graphs obtained from $T_{n,r}$ by adding $q$ edges, and let $λ(G)$ be the spectral radius of a graph $G$. We first prove that for any color-critical graph $F$ with chromatic number $r+1$, there exists $δ_F > 0$ such that for sufficiently large $n$ and all $1 \leq q \leq δ_F \sqrt{n}$, any $n$-vertex graph $G$ with $λ(G) \ge \min_{T \in \mathcal{T}_{n,r,q}} λ(T)$ contains at least $q \cdot c(n,F)$ copies of $F$, where $c(n,F)$ denotes the minimum number of copies of $F$ created by adding a single edge to $T_{n,r}$; moreover, any extremal graph $G$ must belong to $ \mathcal{T}_{n,r,q}$.Next, we prove a spectral supersaturation result for the analogous condition $λ(G) \ge \max_{T \in \mathcal{T}_{n,r,q}} λ(T)$, valid for all $1 \leq q \leq δ_F n$. Together, these results provide a complete resolution to a problem proposed by Ning-Zhai, and establish a spectral counterpart to the well-known results of Mubayi and Pikhurko-Yilma in the extremal supersaturation setting. A notable feature of our first result is that the restriction $q = O(\sqrt{n})$ is tight up to a constant factor, in contrast to the linear bounds provided by other settings discussed above. As applications, we extend a result of Liu-Mubayi, and solve a related conjecture by Li-Lu-Peng.

math.CO↗
arXiv2025

Strong spectral stabilities for $C_{2k+1}$-free graphs

A stability result due to Ren, Wang, Wang and Yang [SIAM J. Discrete Math. 38 (2024)] shows that if $3\le r \le 2k$ and $n\ge 318 (r-2)^2k$, and $G$ is a $C_{2k+1}$-free graph on $n$ vertices with $e(G)\ge \lfloor {(n-r+1)^2}/{4}\rfloor +{r \choose 2}$, then $G$ can be made bipartite by deleting at most $r-2$ vertices. Using a different method, we give a linear bound on $n$ in terms of $k$ and show a stronger structural result, which roughly says that $G$ can be obtained from a large bipartite graph by suspending some small graphs that the total number of vertices is at most $r-2$. This improves a result of Yan and Peng (2024) by weakening the requirement on $n$ and $k$. As a direct corollary, we obtain a tight upper bound on the size of an $n$-vertex $C_{2k+1}$-free graph with chromatic number $χ(G)\ge r$ for every $r\le 2k$. The second part of this paper concerns the spectral extremal problem for $C_{2k+1}$-free graphs. We denote by $λ(G)$ the spectral radius of the adjacency matrix of a graph $G$. Let $T_{n-r+1,2}\circ K_r$ be the graph obtained by identifying a vertex of the complete graph $K_r$ and a vertex of the smaller partite set of the bipartite Turán graph $T_{n-r+1 ,2}$. Using the spectral techniques, we prove that if $3\le r\le 2k$ and $n\ge 712k$, and $G$ is an $n$-vertex $C_{2k+1}$-free graph with chromatic number $χ(G) \ge r$, then $λ(G)\le λ(T_{n-r+1,2}\circ K_r)$, where the equality holds if and only if $G=T_{n-r+1,2}\circ K_r$. Our result not only extends a result of Guo, Lin and Zhao [Linear Algebra Appl. 627 (2021)] as well as a result of Zhang and Zhao [Discrete Math. 346 (2023)], but also provides the first solution to the spectral extremal problem for $F$-free graphs with high chromatic number.

math.CO↗
arXiv2025

An edge-spectral Erdős-Stone-Simonovits theorem and its stability

We study the extremal problem that relates the spectral radius $λ(G)$ of an $F$-free graph $G$ with its number of edges. Firstly, we prove that for any graph $F$ with chromatic number $χ(F)=r+1\ge 3$, if $G$ is an $F$-free graph on $m$ edges, then $λ^2(G)\le {(1-\frac{1}{r} + o(1))2m}$. This provides a unified extension of both the Erdős--Stone--Simonovits theorem and its vertex-spectral version due to Nikiforov, and confirms a conjecture proposed by Li, Liu and Feng. We also establish the corresponding edge-spectral stability, showing that if $G$ is an $F$-free graph on $m$ edges with $λ^2(G)=(1- \frac{1}{r} - o(1))2m$, then $G$ differs from a complete bipartite graph by $o(m)$ edges when $r=2$, and $G$ differs from an $r$-partite Turán graph by $o(m)$ edges when $r\ge 3$. This extends the classical Erdős--Simonovits stability theorem. As an application of our method, we improve a result of Zhai, Lin and Shu by showing that if $λ(G)>\sqrt{m}$, then there exist two vertices in $G$ that have at least $\frac{1}{2}\sqrt{m} - O(1)$ common neighbors. This bound is the best possible as witnessed by a random construction.

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