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Quanyu Tang

Publications and source records attributed to Quanyu Tang.

At least 37 records · Page 2Linked to original sources
arXiv2026

A positive square-energy strengthening of Turán's theorem

Let $G$ be an $n$-vertex graph with clique number $ω(G)$, and let $s^+(G)$ denote the sum of the squared positive adjacency eigenvalues. We prove that $$ \sqrt{s^+(G)}\le\left(1-\frac{1}{ω(G)}\right)n. $$ This strengthens Wilf's classical spectral Turán theorem and resolves a conjecture of Elphick and Wocjan. Adopting the relaxation of our companion paper on the square-energy conjecture, we reduce the theorem to a Motzkin--Straus inequality for doubly nonnegative matrices, which we prove via a local inverse-probability estimate for the Caro--Wei greedy algorithm on the complement.

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

Rational values and non-principality for $\ell$-degree Turán densities of hypergraphs

Let $k>\ell\ge 1$ be integers. For a family $\mathcal{F}$ of $k$-uniform hypergraphs ($k$-graphs), the $\ell$-degree Turán density $γ^{(k)}_\ell(\mathcal{F})$ of $\mathcal{F}$ is defined as the asymptotic maximum of the normalized minimum $\ell$-degree over all $\mathcal{F}$-free $k$-graphs. In this paper, we prove that for all $k>\ell>k/2$, every rational number $α\in[0,1)$ can be realized as the $\ell$-degree Turán density $γ^{(k)}_\ell(\mathcal{F})$ for some finite family $\mathcal{F}$ of $k$-graphs. Furthermore, for any $k>\ell\ge 1$, we construct an explicit infinite sequence of values realized by single forbidden $k$-graphs: for each integer $q\ge2$, there exists a $k$-graph $F$ such that $γ^{(k)}_\ell(F)=1-q^{\ell-k}$. We also establish a strengthened non-principality property for $\ell$-degree Turán densities: for all $k>\ell>1$, there exist two $k$-graphs $F_1$ and $F_2$ satisfying $0< γ^{(k)}_{\ell}(\{F_1,F_2\})<\min\{ γ^{(k)}_\ell(F_1),γ^{(k)}_\ell(F_2)\}$.

math.CO↗
arXiv2026

Generalized spectral closedness of $\mathcal{F}$-free graph classes

In this paper, we investigate the generalized spectral closedness of graph classes defined by a family $\mathcal{F}$ of forbidden induced subgraphs. To systematically study this property, we introduce a novel combinatorial concept of patterned closed walks (or $β$-closed walks), which naturally interlaces the edges of a graph with those of its complement. By establishing the induced-subgraph expansion of these $β$-closed walk counts, we obtain an algebraic sufficient condition for generalized spectral closedness based on the existence of a walk-realizable $\mathcal{F}$-supporter. Crucially, the search for such a walk-realizable supporter is reduced to a linear programming feasibility problem. As primary applications of this computational framework, we prove that the classes of threshold graphs and chain graphs are generalized spectrally closed.

math.CO↗
arXiv2026

Irregular subgraph in a regular graph

A conjecture of Alon and Wei states that, for any $d$-regular graph $G$ with $n$ vertices, there exists a spanning subgraph $H$ such that for all $0\le i\le d$, we have $m(H, i)$, the number of vertices in $H$ with degree $i$, is between $\frac{n}{d+1}-2$ and $\frac{n}{d+1}+2$. We prove the conjecture for all fixed $d$ when $n$ is sufficiently large. More precisely, if $q=(q_0,\ldots,q_d)$ satisfies $$ \sum_{i=0}^d q_i=n,\qquad \sum_{i=0}^d i q_i\equiv 0\pmod 2,\qquad \left|q_i-\frac{n}{d+1}\right|\le 1 \quad (0\le i\le d), $$ then there is a spanning subgraph $H\subseteq G$ such that $$ m(H,i)=q_i \qquad (0\le i\le d). $$

math.CO↗
arXiv2026

Sharp ratios for low-index Neumann eigenvalues on convex domains

Let $Ω\subset\mathbb{R}^N$ be a bounded open convex set, and let $0=μ_0(Ω)<μ_1(Ω)\le μ_2(Ω)\le\cdots$ be the Neumann eigenvalues of the Laplacian, repeated according to multiplicity. We prove the sharp bounds $$ μ_2(Ω)\le 4μ_1(Ω),\qquad μ_3(Ω)\le 9μ_1(Ω). $$ The first estimate resolves a problem attributed to Henrot, while the second gives the next sharp case predicted by the one-dimensional model. The constants are optimal in every dimension.

math.AP↗
arXiv2026

Nordhaus--Gaddum type bounds for the complement rank

Let $G$ be an $n$-vertex simple graph with adjacency matrix $A_G$. The complement rank of $G$ is defined as $\operatorname{rank}(A_G+I)$, where $I$ is the identity matrix. In this paper we study Nordhaus--Gaddum type bounds for the complement rank. We prove that for every graph $G$, $$ \operatorname{rank}(A_G+I)\cdot\operatorname{rank}(A_{\overline G}+I) \ge n, \qquad \operatorname{rank}(A_G+I)+\operatorname{rank}(A_{\overline G}+I) \ge n+1, $$ with the equality cases characterized. We further obtain strengthened multiplicative lower bounds under additional structural assumptions. Finally, we show that the trivial upper bounds $$ \operatorname{rank}(A_G+I)\cdot\operatorname{rank}(A_{\overline G}+I) \le n^2, \qquad \operatorname{rank}(A_G+I)+\operatorname{rank}(A_{\overline G}+I) \le 2n $$ are tight by explicitly constructing, for every $n\ge 4$, graphs $G$ with $\operatorname{rank}(A_G+I)=\operatorname{rank}(A_{\overline G}+I)=n$.

math.CO↗
arXiv2026

Path-Minimality of $p$-Energy for Connected Graphs

Let $G$ be a simple connected graph on $n$ vertices, and let $λ_1(G),λ_2(G),\ldots,λ_n(G)$ be the eigenvalues of its adjacency matrix $A(G)$. For $p>0$, define the $p$-energy of $G$ by $\mathcal E_p(G)=\sum_{i=1}^n |λ_i(G)|^p$. We prove that, for every real number $p\ge 2$ and every simple connected graph $G$ on $n$ vertices, $$ \mathcal E_p(G)\ge \mathcal E_p(P_n), $$ where $P_n$ denotes the path on $n$ vertices. Moreover, for each fixed $p>2$, equality holds if and only if $G\cong P_n$. Together with the previously known star-minimality results, this completes the solution of two questions of Nikiforov. The proof combines two different comparison principles. For $2<p<4$, we use a bipartite reduction, a Mellin representation of fractional powers, and a determinant comparison involving matching generating polynomials and tree shifts. For $p\ge4$, we prove a second-order stop-loss comparison for the squared singular values of bipartite graphs. This comparison is established by rank-one spectral-shift estimates, deletion-minimal counterexamples, and a finite certified analysis of the terminal sparse-sun configurations.

math.CO↗
arXiv2026

The Ashbaugh--Benguria reciprocal-gap conjecture for Dirichlet eigenvalues

We prove the Ashbaugh--Benguria reciprocal-gap conjecture for the Dirichlet Laplacian in every dimension $N\ge2$. Specifically, if $Ω\subset\mathbb R^N$ is a bounded domain and $$ 0<λ_1(Ω)<λ_2(Ω)\leλ_3(Ω)\le\cdots $$ are its Dirichlet eigenvalues, then $$ \sum_{i=1}^{N} \frac{λ_1(Ω)} {λ_{i+1}(Ω)-λ_1(Ω)} \ge \frac{N}{j_{N/2,1}^2/j_{N/2-1,1}^2-1}, $$ where $j_{μ,1}$ denotes the first positive zero of the Bessel function $J_μ$ of the first kind of order $μ$. We also characterize the equality case: equality holds precisely when $Ω$ agrees with a Euclidean ball up to a set of Sobolev $H^1$-capacity zero. In particular, among bounded Lipschitz domains, equality holds if and only if $Ω$ is a Euclidean ball.

math.AP↗
arXiv2026

Path-Minimality for Positive $p$-Energies, Laplacian-Type Spectra, and Line Graphs

We derive several applications of the path-minimality theorem for adjacency $p$-energy proved in the companion paper. First, we prove the sharp inequality $$ \mathcal E_p^+(G)\ge \mathcal E_p^+(P_n), $$ where $P_n$ is the path on $n$ vertices, in three settings: connected bipartite graphs for every real $p\ge2$, all connected graphs for every odd integer $p\ge3$, and all connected graphs for $p=4$. Second, using subdivision graphs, we prove path-minimality for Laplacian and signless Laplacian-type spectral sums, including power sums, Estrada-type quantities, resolvent energies, and thresholded tails. Third, we prove an edge-count second-order stop-loss comparison for the signless Laplacian above the threshold $2$. This yields the sharp line-graph inequality $$ \mathcal E_p^+(\mathcal L(G))\ge \mathcal E_p^+(P_m) $$ for every connected graph $G$ with $m$ edges and every real $p\ge2$.

math.CO↗
arXiv2026

A sharp fixed-volume product inequality for the first $N$ nonzero Steklov eigenvalues

We prove a sharp fixed-volume product inequality for the first $N$ nonzero Steklov eigenvalues of bounded Lipschitz domains in $\mathbb R^N$. More precisely, if $N\ge2$ and $Ω\subset\mathbb R^N$ is a bounded Lipschitz domain, then $$ \prod_{j=1}^N σ_j(Ω)\le \frac{ω_N}{|Ω|}, $$ where $0=σ_0(Ω)<σ_1(Ω)\leσ_2(Ω)\le\cdots$ are the Steklov eigenvalues of $Ω$, and $ω_N$ denotes the volume of the unit ball in $\mathbb R^N$. This extends the convex-domain theorem of Henrot, Philippin, and Safoui to arbitrary bounded Lipschitz domains, and in particular settles the remaining higher-dimensional case of a problem posed by Henrot.

math.AP↗
arXiv2026

Consecutive integers free of certain prime factors

Let $n_k$ denote the least integer $n>2k$ such that $(n-k)(n-k+1)\cdots(n-1)$ is not divisible by any prime in the interval $(k,2k)$. Confirming a conjecture of Erdős, we prove that, for all sufficiently large $k$, $$ n_k > e^{\frac{\log^2 k}{20 \log \log k}}. $$

math.NT↗
arXiv2026

Gaps in Multiplicative Sidon Sets II

With $ρ= \frac{13-\sqrt{69}}{10} \approx 0.47$, it was recently established that there exist multiplicative Sidon sets (sets without any non-trivial solutions to $ab = cd$) in $\{1, 2, \ldots, n\}$ with maximal gap size $\ll_{\varepsilon} n^{ρ+ \varepsilon}$. Here we improve upon this result and show that one can take $ρ= \frac{10}{33} \approx 0.303$ instead.

math.NT↗
arXiv2026

Proof of a conjectured spectral upper bound on the chromatic number of a graph

Let $G$ be a simple graph on $n$ vertices and $m$ edges with chromatic number $χ$, and let $λ_n$ denote the least adjacency eigenvalue. Solving a conjecture of Fan, Yu and Wang~[Electron. J. Combin., 2012], we prove that when $3\le χ\le n-1$, the chromatic number satisfies the following upper bound: $$ χ\le \left(\frac{n}{2}+1+λ_n\right) + \sqrt{\left(\frac{n}{2}+1+λ_n\right)^{2}-4(λ_n+1)\left(λ_n+\frac{n}{2}\right)}, $$ with equality if and only if $G \cong \left(K_{\fracχ{2}}\cup\tfrac{n-χ}{2}K_1\right) \vee \left(K_{\fracχ{2}}\cup\tfrac{n-χ}{2}K_1\right)$, where both $n$ and $χ$ are even. This extends the validity of Fan--Yu--Wang's bound from the range $3\le χ\le \frac{n}{2}$ to the full range $3\le χ\le n-1$. We also compare this bound with the well-known bound due to Wilf that $χ\le 1 + λ_1$, where $λ_1$ denotes the largest eigenvalue. In particular we show that while Wilf's bound is an upper bound for some parameters larger than $χ$, this bound using $λ_n$ is not an upper bound for these parameters. We conclude with a similar conjectured upper bound for $χ(G)$, which uses $m$ in place of $n$.

math.CO↗
arXiv2026

The separable case of Kadison's problem on orthonormal bases of unitaries for type $\mathrm{II}_1$ factors

In 1967, Kadison asked ``does every type $\mathrm{II}_1$ factor have an orthonormal (with respect to the trace) basis consisting of unitaries?'' Using a noncommutative Lyapunov theorem of Akemann and Weaver, we prove that if $M$ is a separable diffuse finite von Neumann algebra with a normal faithful trace $τ$, then $L^2(M,τ)$ admits an orthonormal basis consisting of self-adjoint unitaries in $M$. Consequently, we affirm the separable case of the Kadison problem.

math.OA↗
arXiv2026

Almost Affine Invariance Over Prime Fields: Green Problem 90

Let $A\subset \mathbb{F}_p$ with density 1/2. We call a set $A$ almost affine invariant under an affine transformation $ϕ(x)=ax+b$ if \[|A \triangle ϕ(A)| =o(p).\] We determine that, the threshold value of $K$ such that $A$ is almost affine invariant simultaneously under all $ϕ(x)$ with $|a|, |b|\le K$ and $a\neq 0$, is $K=o(\log p)$. This solves Ben Green's Open Problem 90.

math.CO↗
arXiv2026

Factorization of invariant polynomials and generalized spectral characterizations of graphs

The problem of characterizing graphs by their generalized spectra has received significant attention in recent years. This paper provides a complete proof of a conjecture proposed by Wang, Wang, and Zhu (European J. Combin., 2023), which asserts that the square-root polynomial of the invariant polynomial $Φ_p(G;x) \in \mathbb{F}_p[x]$ can replace its square-free part to yield a more effective criterion for a graph to be determined by its generalized spectrum (DGS). A key ingredient of our proof is a novel algebraic factorization: we show that the polynomial $Φ_p(G;x)$ is the product of the characteristic polynomials of the adjacency operator restricted to the left null space of the walk matrix and its radical, respectively. Based on this refined DGS-criterion, a broad family of DGS-graphs is constructed via rooted products, significantly generalizing the recent result of Wang, Shen, and Mao (Discrete Appl. Math., 2026).

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