arXiv subjects

Yong Lu

Publications and source records attributed to Yong Lu.

At least 19 recordsLinked to original sources

arXiv2026

On the Turán number for the Cartesian product of an odd cycle and a tree

Let $T$ be a fixed tree on at least two vertices. We determine the Turán number and all extremal graphs for the Cartesian product of an odd cycle $C_{2m+1}$ and $T$, for every fixed integer $m\ge1$ and all sufficiently large $n$, in terms of its decomposition family. Every extremal graph is the join of an independent set and an extremal graph for that family. The proof combines the Erdős--Sós theorem with weak stability and progressive induction.

math.CO↗

arXiv2026

Triangle-free Rank--Ramsey graphs from ordered incidence lifts

For a graph $G$ with adjacency matrix $A_G$ and a coefficient field $\mathbb K$, put $ρ_{\mathbb K}(G)=\operatorname{rank}_{\mathbb K}(A_G+I)$. We answer the triangle-free Rank--Ramsey existence question of Beniamini, Linial and Shraibman. For every order $n$, we construct a triangle-free graph $H_n$ such that, simultaneously over all coefficient fields, $$ ρ_{\mathbb K}(H_n)=\bigl(4(2/3)^{3/4}+o(1)\bigr)n^{3/4}, $$ with an error uniform in the field. The graphs can be chosen Hamiltonian for all sufficiently large $n$. Our method combines ordered layers of point--line incidence flags with a common low-dimensional endpoint space. We determine the exact real rank, independence number and chromatic number of the affine lifts. For all triangle-free complete incidence lifts, we obtain a sharp two-term lower bound for complete incidence lifts, with sharpness witnessed by projective-plane incidence graphs over suitable coefficient fields. We also establish a third-order estimate and stability near equality. On the same explicit graphs, both the Lovász number and minimum semidefinite rank exceed the prescribed Boolean-matrix rank by polynomial factors. Thus neither parameter has a universal constant-factor upper bound in terms of complement rank. We determine Shannon capacity in the short-layer range.

math.CO↗

arXiv2026

Strict spectral supersaturation for cliques: extremal graphs and sharp thresholds

For every fixed $r\ge3$ and all sufficiently large $n$, we determine the largest adjacency spectral radius of an $n$-vertex graph with fewer than $q c_r(n)$ copies of $K_{r+1}$, where $1\le q<n/r$. Here $c_r(n)$ is the number of copies created by adding one edge to a largest part of the Turán graph. We also determine all extremal graphs. In most cases the extremal graph is obtained from an almost balanced complete multipartite graph by adding a star in one part. Two small values of $q$ require separate constructions, and an additional transition occurs when $r=3$ and $n\equiv2\pmod3$. Under the non-strict constraint, the unique extremal graph is obtained by adding a $q$-edge star to a largest part of the Turán graph. We determine the difference between the strict and non-strict values and prove that the sharp matching threshold is $\sqrt2(r-1)/r$. The proof treats separately the ranges $q=o(n)$, $q/m\toγ\in(0,1)$, and $q/m\to1$.

math.CO↗

arXiv2026

A counterexample to the Fang--Lin conjecture: Edge and spectral extremality diverge near a Turán graph

Fang and Lin [J. Algebraic Combin. 63 (2026), Art.~58] asked whether, whenever $F$ is edge-color-critical with $χ(F)=r+1$, every non-$r$-partite, $F$-free graph of maximum adjacency spectral radius must also maximize the number of edges. We give a negative answer. Let $F=K_1\veeμ(K_3)$, where $μ(K_3)$ is the Mycielskian of a triangle. This graph is edge-color-critical with $χ(F)=5$. We prove that $\SPEX_{5}(n,F)\cap\EX_{5}(n,F)=\varnothing$ for all sufficiently large $n$. Thus no graph can simultaneously maximize both the edge count and the spectral radius.

math.CO↗

arXiv2026

Quantum-hardware spectral co-design framework for multifrequency Rydberg electrometry

Engineering electromagnetic hardware to satisfy discrete quantum-defined spectral constraints constitutes a general spectral co-design problem for quantum systems. Here we address this challenge in multifrequency Rydberg electrometry by directly coupling a fabrication-constrained simultaneous perturbation stochastic approximation (SPSA)--Adam optimizer to full-wave finite-element eigenmode simulations. Requiring neither analytical nor adjoint gradients, the method operates over a discrete design space containing approximately $10^{300}$-configurations and yields a novel multimode electrometry architecture that simultaneously aligns four high-$Q$ eigenmodes with four selected Cs Rydberg transitions. The optimized design remains highly robust to fabrication imperfections, achieving a relative frequency error as low as $8.42\times10^{-7}$ while reducing the device length by a factor of $1.75\times10^{2}$, thereby overcoming the difficulty of simultaneous multimode spectral matching encountered in conventional design. For a comparable simulation budget, the proposed co-design framework achieves frequency-matching errors approximately 10 and 41 times smaller than those of covariance matrix adaptation evolution strategy and discrete simulated annealing, respectively. The electrometry is predicted to provide an average input-power-sensitivity enhancement of approximately $4.04\times10^{3}$, demonstrating quantum--hardware spectral co-design as a general route toward compact hardware for multichannel quantum sensing.

quant-ph↗

arXiv2026

A cardinal trichotomy for topological equivalence classes of countable trees

For a tree $T$, let $[T]$ be the set of isomorphism classes of trees that are mutually topological minors of $T$. Bruno and Szeptycki proved that every locally finite tree satisfies $|[T]|\in\{1,2^{\aleph_0}\}$ and that every tree with a ray containing infinitely many vertices of degree at least $3$ has at least $2^{\aleph_0}$ topological twins. Hence equality holds in the latter case when the tree is countable. We treat the complementary countable case, without any bound on the degrees. For every such tree $T$, we prove that $|[T]|\in\{1,\aleph_0,2^{\aleph_0}\}$ and give a well-founded recursive description based on Schmidt rank. The proof uses a finite canonical subtree extracted from the subtree generated by the branching vertices and a counting theorem for countable multisets over a well-quasi-order. The same trichotomy follows for every countable tree and hence for every tree all of whose vertices have countable degree.

math.LO↗

arXiv2026

Bounded-Defect Proper Orientations and Graphs on Surfaces

Chen, Mohar and Wu [J. Combin. Theory Ser. B 161 (2023)] conjectured that every graph of orientable genus $g$ has proper orientation number $O(\sqrt g)$. We prove this conjecture. Our main result extends the potential-outdegree method from independent color classes to three parts of bounded internal degree. Let $k,D\in\mathbb Z_{\ge0}$. If $G$ has maximum average degree at most $2k$ and $V(G)=V_1\cup V_2\cup V_3$, where each $V_i$ induces a subgraph of maximum degree at most $D$, then $$\pchi(G)\le k+9D+6+\floor{\frac32\ceil{\frac{2D+2}{5}}}\le k+\floor{\frac{48D}{5}}+8.$$ When the partition is given, we construct such an orientation in polynomial time. For every $D\ge1$, we also construct a graph with a partition into three parts of internal maximum degree at most $D$ for which $\pchi(G)-\ceil{\mad(G)/2}\ge6D+5$; hence the linear dependence on $D$ is unavoidable. Finally, Woodall's defective-coloring theorem yields $\pchi(G)=O(\sqrt{γ+1})$ for every graph of Euler genus $γ$.

math.CO↗

arXiv2026

Exact random covers of metric trees: balanced rounding, duality, and sharp thresholds

Norin and Turcotte's asymptotically sharp bound for graph burning [J. Combin. Theory Ser. B 168 (2024), 208--235] led them to an exact random-cover conjecture for finite metric trees. Let $U[0,r]$ be the uniform probability measure on $[0,r]$. They conjectured that every finite metric tree $T$ of length $L\ge2r$ admits a probability measure on $0$-good ball covers whose expected radius measure is at most $(L/r)U[0,r]$. We prove the conjecture for every finite metric tree. We recast the bootstrapping calculation of Norin and Turcotte as a zero-error replacement certificate. The resulting local scale reduction, together with a three-piece decomposition and a macro-recursion, produces a fractional marked-ball cover with the exact radius budget. We then pass from the fractional cover to random finite covers by a compact rounding argument. For metric-tree balls, Tamir's balancedness theorem and standard balanced-matrix ideality provide the finite-dimensional integrality input. We also prove an arbitrary-budget duality criterion. If $0<R\le L$ and $β$ is a finite positive Borel measure on $[0,R]$, then $β$ dominates the expected radius measure of a random $0$-good cover if and only if $σ(T)\le\int_{[0,R]}\max_{v\in T}σ(B_T(v,s))\,dβ(s)$ for every finite positive Borel measure $σ$ on $T$; it is enough to test finite atomic measures. We use this criterion to extend the uniform range to every $r\le L-\operatorname{diam}(T)/2$, determine the exact range for equal-arm metric stars, and derive deterministic bounds, interval rigidity, and a diameter-defect stability estimate.

math.CO↗

arXiv2026

A Proof of the Chen--Raspaud Conjecture

For every integer $k\ge2$, Chen and Raspaud conjectured that each graph $G$ with odd girth $\og(G)\ge2k+1$ and maximum average degree $\mad(G)<2+1/k$ has a $(2k+1:k)$-coloring. In this paper, we prove the conjecture.

math.CO↗

arXiv2026

Optimal convergence rates of the Klein-Gordon-Schrödinger system in the nonrelativistic limit

In this paper, we study the Klein-Gordon-Schrödinger system in the nonrelativistic regime $ε\to 0$, where $ε$ is proportional to the inverse of the speed of light. We show that the Klein-Gordon-Schrödinger system converges to a system of decoupled linear Schrödinger equations over a long time interval of order $ε^{-1}$ with error estimates of the form $(1+t)ε^2$; in particular, the error estimate for the Schrödinger component is uniform in time of the form $ε^{2}$ The specific forms of the error estimates coincide with the numerical results shown by Bao et a.l., and the $O(ε^{2})$ convergence rates coincide with the order of initial error, and thus are optimal.

math.AP↗

arXiv2026

Hamiltonian cycles in 7-tough $(P_4\cup P_1)$-free graphs

Shan~[J. Graph Theory (2026)] proved that every 23-tough $(P_4\cup P_1)$-free graph on at least three vertices is Hamiltonian. We improve this bound to 7 by replacing the final cut analysis in Shan's framework with an asymmetric separation criterion and a cograph covering lemma.

math.CO↗

arXiv2026

Homogenization of the Navier-Stokes-Cahn-Hilliard system in the small-hole regime

This paper investigates the homogenization of the 3D Navier--Stokes--Cahn--Hilliard (NSCH) system in domains containing a large number of solid obstacles (named holes). Each hole has diameter of order $\varepsilon^α(α>3)$, where $\varepsilon > 0$ denotes the small length scale for inter-hole separation. Both viscosity and mobility depend on the phase-field variable. We establish two distinct asymptotic regimes: if the capillary strength $λ_\varepsilon\to λ>0$ as $\varepsilon\to 0$, the limit system coincides with the original NSCH system; if $λ_\varepsilon\to 0$ as $\varepsilon\to 0$, the scaled velocity, phase field and chemical potential converge to a weak solution to a Stokes--Cahn--Hilliard (SCH) system. To the best of our knowledge, this work constitutes the first rigorous homogenization analysis for evolutionary NSCH flows with phase-dependent viscosity and mobility under the subcritical hole scaling.

math.AP↗

arXiv2026

Homogenization of a non-homogeneous incompressible heat-conducting fluid in perforated domains

This paper provides the study of the homogenization of the 3D non-homogeneous incompressible heat-conducting fluid in perforated domains with holes of subcritical size, where the viscosity and the heat conductivity coefficient are assumed to depend on the temperature. The diameter of the holes is of order $\varepsilon^α \ (α>3)$, where $\varepsilon > 0$ is a small parameter that measures the mutual distance between the holes. We prove that as $\varepsilon\to 0$, the limit behavior of velocity, density and temperature is governed by the original system in homogeneous domain without holes.

math.AP↗

arXiv2026

Qualitative and quantitative homogenization of some non-Newtonian flows in perforated domains: case of `small holes'

We consider the homogenization of three dimensional viscous incompressible non-Newtonian flows satisfying certain general $r$-structure in perforated domains. We focus on the case of `small holes' by assuming the holes under consideration are of size $\varepsilon^α$ with $α>3$, where $\varepsilon$ is the perforation parameter used to measure the mutual distance between the holes. We show the limit equations remain unchanged in the homogenization limit under the constraint $ \frac{6(α- 1)}{4α-5}< r<3-\frac{3}α$, which seems optimal in the sense of Sobolev capacity of holes as explained in Remark 1.3. Quantitative convergence rates are further derived for both the velocity field and the pressure. To the best of our knowledge, both the qualitative and quantitative homogenization results are firstly given for non-Newtonian flows in the `small holes' case.

math.AP↗

arXiv2026

Sufficient conditions for fractional $k$-factor-critical graphs with minimum degree to be $k$-factor-critical

A graph $G$ is called $k$-factor-critical if after deleting any $k$ vertices the remaining subgraph still has a perfect matching. Fan and Lin [Adv. in Appl. Math. 174 (2026) 103019] posed an adjacency spectral condition for a graph with minimum degree to be $k$-factor-critical. A graph $G$ is fractional $k$-factor-critical if after deleting any $k$ vertices the remaining subgraph still has a fractional perfect matching. Clearly, the fractional $k$-factor-criticality of a graph is a necessary property for a graph to be $k$-factor-critical. Jia, Fan and Liu [Discrete Appl. Math. 386 (2026) 255-263] proposed a tight sufficient condition in terms of the spectral radius for a graph with fractional $k$-factor-criticality to be $k$-factor-critical. A natural question arises: can we derive analogous sufficient conditions by incorporating the minimum degree parameter of graphs? We first establish a lower bound on the size to ensure that a $(k+1)$-connected graph with fractional $k$-factor-criticality is $k$-factor-critical, where $k$ is a positive integer with $k\geq1$. Moreover, we provide a sufficient condition in terms of the spectral radius for a $(k+1)$-connected graph with fractional $k$-factor-criticality to be $k$-factor-critical. Our results generalize the result of Jia, Fan and Liu to $(k+1)$-connected graphs. Furthermore, our spectral conditions apply to a broader family of connected graphs compared with the results of Fan and Lin, as well as Jia et al.

math.CO↗

arXiv2026

Correlation Matrix Method for Phonon Quasiparticles

Phonon anharmonicity is ubiquitous in real materials and is crucial for understanding thermal properties and phase stability. In this work, we show that anharmonic phonon modes can be obtained by maximizing their vibration stability during fitting the atomic trajectory. We prove that all information about these quasiparticles is contained in two small correlation matrices $\mathcal{S}$ and $\mathcal{Q}$, which can be constructed directly from molecular dynamics simulations. Based on these matrices, we proposed an optimization scheme, which allows us to efficiently determine temperature-dependent phonon modes along with their frequencies and lifetimes. We verified this method by applying it to silicon and cubic CaSiO$_3$, where it successfully captured their temperature-dependent phonon behaviors and the well-known phonon softening in cubic CaSiO$_3$. This theory provides a convenient tool for investigating phonon quasiparticles and can be extended to study other quasiparticles, such as electrons, holes, and magnons.

cond-mat.mtrl-sci↗

arXiv2026

Sufficient conditions for spanning $k$-trees in tough graphs

The toughness of a graph $G$, denoted by $τ(G)$, is defined by $τ(G)=$min $\{\frac{|S|}{c(G-S)}:S\subseteq V(G)$ and $c(G-S)\geq2\}$. A graph $G$ is said to be $τ$-tough if $τ(G)\geq τ$. Let $k\geq2$ be an integer. A tree $T$ is called a $k$-tree if $d_{T}(v)\leq k$ for each $v\in V(T)$, that is, the maximum degree of a $k$-tree is at most $k$. A $k$-tree $T$ is a spanning $k$-tree if $T$ is a spanning subgraph of a connected graph $G$. In 1989, Win [Graphs Combin. 5 (1989) 201--205] proved that if $τ(G)\geq\frac{1}{k-2}$, where $k\geq3$, then $G$ contains a spanning $k$-tree. Liu, Fan and Shu [Discrete Math. 348 (2025) 114593] provided a tight sufficient condition based on the spectral condition for connected $\frac{1}{k}$-tough and $\frac{1}{k-1}$-tough graphs to contain a spanning $k$-tree, where $k\geq3$ is an integer. A natural and interesting problem arises: Can the value of $τ$ be refined? When $\frac{1}{k-2}>τ\geq\frac{1}{k-1}$, we initially establish a lower bound on the size to ensure that a connected $\frac{t}{t(k-2)+1}$-tough graph $G$ contains a spanning $k$-tree, where $k\geq3$ and $t\geq1$ are integers. Meanwhile, we provide two sufficient conditions in terms of spectral radius and signless Laplacian spectral radius for a connected $\frac{t}{t(k-2)+1}$-tough graph $G$ to contain a spanning $k$-tree, where $k\geq3$ and $t\geq1$ are integers. When $t=1$, we obtain the result $η=1$ from Liu, Fan and Shu.

math.CO↗

arXiv2026

Some sufficient conditions for a graph with minimum degree to be $k$-critical with respect to $[1,b]$-odd factors

A graph $G$ is $k$-factor-critical if $G-S$ has a perfect matching for every subset $S \subseteq V(G)$ with $|S|=k$. A spanning subgraph $H$ of $G$ is called a $[1,b]$-odd factor if $b \equiv 1 \pmod{2}$ and $d_{H}(v) \in\left\lbrace 1, 3, \ldots, b\right\rbrace$ for every $v\in V(G),$ where $d_{H}(v)$ denotes the degree of vertex $v$ in $H$. Moreover, $G$ is said to be $k$-critical with respect to $[1,b]$-odd factors if $G-X$ contains a $[1,b]$-odd factor for every subset $X \subseteq V(G)$ with $|X|=k$. In this paper, we provide some sufficient conditions based on the distance spectral radius and the distance signless Laplacian spectral radius for a graph with minimum degree to be $k$-critical with respect to $[1,b]$-odd factors.

math.CO↗