arXiv subjects

Xiao-Dong Zhang

Publications and source records attributed to Xiao-Dong Zhang.

At least 37 records · Page 2Linked to original sources

arXiv2025

Exact Turán numbers of two vertex-disjoint paths

The Turán number of a graph $H$ is the maximum number of edges in any graph of order $n$ that does not contain $H$ as a subgraph. In 1959, Erd\H os and Gallai obtained a sharp upper bound of Turán numbers for a path of arbitrary length. In 1975, Faudree and Schelp, and independently in 1977, Kopylov determined the exact values of Turán numbers of paths with arbitrary length. In this paper, we determine the Turán number of two vertex-disjoint paths of odd order at least 4. Together with previous works, we determine the exact Turán numbers of two vertex-disjoint paths completely. This confirms the first $k=2$ case of a conjecture proposed by Yuan and Zhang in 2021, which generalizes the Turán number formula of paths due to Faudree-Schelp, and Kopylov in a broader setting. Our main tools include a refinement of Pósa's rotation lemma, a stability result of Kopylov's theorem on cycles, and a recent inequality on circumference, minimum degree, and clique number of a 2-connected graph.

math.CO↗

arXiv2025

Quantum Algorithms for Solving Generalized Linear Systems via Momentum Accelerated Gradient and Schrödingerization

In this paper, we propose a quantum algorithm that combines the momentum accelerated gradient method with Schrödingerization [S. Jin, N. Liu and Y. Yu, Phys. Rev. Lett, 133 (2024), 230602][S. Jin, N. Liu and Y. Yu, Phys. Rev. A, 108 (2023), 032603], achieving polynomial speedup over its classical counterpark in solving linear systems. The algorithm achieves a query complexity of the same order as the Schrödingerization based damped dynamical system method, namely, linear dependence on the condition number of the matrix, and can overcome the practical limitations of existing non-Schrödingerization-based quantum linear system algorithms. These limitations stem from their reliance on techniques such as VTAA and RM, which introduce substantial quantum hardware resource overhead. Furthermore, it demonstrates both theoretically and experimentally that the auxiliary variables introduced by our method do not dominate the error reduction at any point, thereby preventing a significant increase in the actual evolution time compared to the theoretical prediction. In contrast, the damped method fails to meet this criterion. This gives new perspectives for quantum algorithms for linear systems, establishing a novel analytical framework for algorithms with broader applicability, faster convergence rates, and superior solution quality.

quant-ph↗

arXiv2025

Generalized Turan number with given size

Generalized Turán problem with given size, denoted as $\mathrm{mex}(m,K_r,F)$, determines the maximum number of $K_r$-copies in an $F$-free graph with $m$ edges. We prove that for $r\ge 3$ and $α\in(\frac 2 r,1]$, any graph $G$ with $m$ edges and $Ω(m^{\frac{αr}{2}})$ $K_r$-copies has a subgraph of order $n_0=Ω(m^\fracα{2})$, which contains $Ω(n_0^{\frac{i(r-2)α}{(2-α)r-2}})$ $K_i$-copies for each $i = 2, \ldots, r$. This implies an upper bound of $\mathrm{mex}(m, K_r, F)$ when an upper bound of $\mathrm{ex}(n,K_r,F)$ is known. Furthermore, we establish an improved upper bound of $\mathrm{mex}(m, K_r, F)$ by $\mathrm{ex}(n, F)$ and $\min_{v_0 \in V(F)} \mathrm{ex}(n, K_r, F - v_0)$. As a corollary, we show $\mathrm{mex}(m, K_r, K_{s,t}) = Θ( m^{\frac{rs - \binom{r}{2}}{2s-1}} )$ for $r \geq 3$, $s \geq 2r-2$ and $t \geq (s-1)! + 1$, and obtain non-trivial bounds for other graph classes such as complete $r$-partite graphs and $K_s \vee C_\ell$, etc.

math.CO↗

arXiv2025

A Method to Decipher "Genome" from Interatomic Cohesion in the Exploration for a "Central Dogma" Replacement in Material Science

In the ball-stick model, interatomic cohesions are considered "sticks". But enormous details and features of the "sticks" are usually oversimplified as indexed quantities or equivocated as geometry characteristics. These indexed quantities or geometry characteristics not only limit the explanatory capability to a few chemical/physical aspects but also eliminate generativity for expected resemblance. And these limitations can be related to the information loss during the conversion. Herein, inspired by the central dogma, a framework is introduced to compact interatomic cohesions into a detailed residue-by-residue "genome" with matched encoding/decoding tools. The framework fuses the quantum mechanical aspects, auto feature extraction, nanostructures and/or simulations, and generative models. As a proof of concept, the realization introduced in this work adopted bosonic/fermionic features, an autoencoder with image recognition processes, Density Functional Theory simulations, and a thiolate-protected gold nanocluster dataset. After repetitive modeling, validating, and analysis based on 26,528 simulated interatomic images, the interatomic cohesion can be almost losslessly encoded into an 8-value-genome, and the genome encoder-decoder pair is also obtained. The model is then automatically extended into a generative model which converts any arbitrary 8-value-genome to a bond image.

cond-mat.mes-hall↗

arXiv2024

Dense halves in balanced 2-partition of K4-free graphs

A balanced 2-partition of a graph is a bipartition $A,A^c$ of $V(G)$ such that $|A|=|A^c|$. Balogh, Clemen, and Lidický conjectured that for every $K_4$-free graph on $n$ (even) vertices, there exists a balanced 2-partition $A,A^c$ such that $\max\{e(A),e(A^c)\}\leq n^2/16$ edges. In this paper, we present a family of counterexamples to the conjecture and provide a new upper bound ($0.074n^2$) for every sufficiently large even integer $n$.

math.CO↗

arXiv2024

Multi-view clustering integrating anchor attribute and structural information

Multisource data has spurred the development of advanced clustering algorithms, such as multi-view clustering, which critically relies on constructing similarity matrices. Traditional algorithms typically generate these matrices from sample attributes alone. However, real-world networks often include pairwise directed topological structures critical for clustering. This paper introduces a novel multi-view clustering algorithm, AAS. It utilizes a two-step proximity approach via anchors in each view, integrating attribute and directed structural information. This approach enhances the clarity of category characteristics in the similarity matrices. The anchor structural similarity matrix leverages strongly connected components of directed graphs. The entire process-from similarity matrices construction to clustering - is consolidated into a unified optimization framework. Comparative experiments on the modified Attribute SBM dataset against eight algorithms affirm the effectiveness and superiority of AAS.

cs.LG↗

arXiv2024

Key motifs searching in complex dynamical systems

Key network motifs searching in complex networks is one of the crucial aspects of network analysis. There has been a series of insightful findings and valuable applications for various scenarios through the analysis of network structures. However, in dynamic systems, slight changes in the choice of dynamic equations and parameters can alter the significance of motifs. The known methods are insufficient to address this issue effectively. In this paper, we introduce a concept of perturbation energy based on the system's Jacobian matrix, and define motif centrality for dynamic systems by seamlessly integrating network topology with dynamic equations. Through simulations, we observe that the key motifs obtained by the proposed energy method present better effective and accurate than them by integrating network topology methods, without significantly increasing algorithm complexity. The finding of key motifs can be used to apply for system control, such as formulating containment policies for the spread of epidemics and protecting fragile ecosystems. Additionally, it makes substantial contribution to a deeper understanding of concepts in physics, such as signal propagation and system's stability.

physics.soc-ph↗

arXiv2024

Localized Version of Hypergraph Erdos-Gallai Theorem

This paper focuses on extensions of the classic Erdős-Gallai Theorem for the set of weighted function of each edge in a graph. The weighted function of an edge $e$ of an $n$-vertex uniform hypergraph $\mathcal{H}$ is defined to a special function with respect to the number of edges of the longest Berge path containing $e$. We prove that the summation of the weighted function of all edges is at most $n$ for an $n$-vertex uniform hypergraph $\mathcal{H}$ and characterize all extremal hypergraphs that attain the value, which strengthens and extends the hypergraph version of the classic Erdős-Gallai Theorem.

math.CO↗

arXiv2024

A class of trees determined by their chromatic symmetric functions

Stanley introduced the concept of chromatic symmetric functions of graphs which extends and refines the notion of chromatic polynomials of graphs, and asked whether trees are determined up to isomorphism by their chromatic symmetric functions. Using the technique of differentiation with respect to power-sum symmetric functions, we give a positive answer to Stanley's question for the class of trees with exactly two vertices of degree at least 3. In addition, we prove that for any tree $T$, the generalized degree sequence for subtrees of $T$ is determined by the chromatic symmetric function of $T$, providing evidence to a conjecture of Crew.

math.CO↗

arXiv2023

Quantum multigraph states and multihypergraph states

We proposed two classes of multiparticle entangled states, the multigraph states and multihypergraph states, defined by unique operations on the edges and hyperedges. A key discovery is the one-to-one correspondence between the proposed multihypergraph states and the generalized real equally weighted states when d is prime. While for composite d, multihypergraph states form a subset of the generalized real equally weighted states. Meanwhile, we detailed a method for constructing real equally weighted states from hypergraph states and revealed the generalized real equally weighted states which cannot be generated from d-dimensional hypergraph states.

quant-ph↗

arXiv2023

The spectral radius of minor free graphs

In this paper, we present a sharp upper bound for the spectral radius of an $n$-vertex graph without $F$-minor for sufficient large $n$, where $F$ is obtained from the complete graph $K_r$ by deleting disjointed paths. Furthermore, the graphs which achieved the sharp bound are characterized. This result may be regarded to be an extended revision of the number of edges in an $n$-vertex graph without $F$-minor.

math.CO↗

arXiv2023

A note on median eigenvalues of subcubic graphs

Let $G$ be an simple graph of order $n$ whose adjacency eigenvalues are $λ_1\ge\dots\geλ_n$. The HL--index of $G$ is defined to be $R(G)= \max\{|λ_{h}|, |λ_{l}|\}$ with $h=\left\lfloor\frac{n+1}{2}\right\rfloor$ and $ l=\left\lceil\frac{n+1}{2}\right\rceil.$ Mohar conjectured that $R(G)\le 1$ for every planar subcubic graph $G$. In this note, we prove that Mohar's Conjecture holds for every $K_4$-minor-free subcubic graph. Note that a $K_4$-minor-free graph is also called a series--parallel graph. In addition, $R(G)\le 1$ for every subcubic graph $G$ which contains a subgraph $K_{2,3}$.

math.CO↗

arXiv2023

Fundamental Patterns of Signal Propagation in Complex Networks

Various disasters stem from minor perturbations, such as the spread of infectious diseases, cascading failure in power grids, etc. Analyzing perturbations is crucial for both theoretical and application fields. Previous researchers have proposed basic propagation patterns for perturbation and explored the impact of basic network motifs on the collective response to these perturbations, However, the current framework is limited in its ability to decouple interactions, and therefore cannot analyze more complex structures. In this article, we establish an effective, robust and powerful propagation framework under a general dynamic model. This framework reveals common and dense network motifs that exert a critical influence on signal propagation, often spanning orders of magnitude compared with conclusions generated by previous work. Moreover, our framework provides a new approach to understand the fundamental principles of complex systems and the negative feedback mechanism, which is of great significance for research of system controlling and network resilience.

physics.soc-ph↗

arXiv2023

Coherent control of a high-orbital hole in a semiconductor quantum dot

Coherently driven semiconductor quantum dots are one of the most promising platforms for non-classical light sources and quantum logic gates which form the foundation of photonic quantum technologies. However, to date, coherent manipulation of single charge carriers in quantum dots is limited mainly to their lowest orbital states. Ultrafast coherent control of high-orbital states is obstructed by the demand for tunable terahertz pulses. To break this constraint, we demonstrate an all-optical method to control high-orbital states of a hole via stimulated Auger process. The coherent nature of the Auger process is proved by Rabi oscillation and Ramsey interference. Harnessing this coherence further enables the investigation of single-hole relaxation mechanism. A hole relaxation time of 161 ps is observed and attributed to the phonon bottleneck effect. Our work opens new possibilities for understanding the fundamental properties of high-orbital states in quantum emitters and developing new types of orbital-based quantum photonic devices.

quant-ph↗

arXiv2023

Spin Cooperated Catalytic Activities in Mn-N4 based Single-atom Nanozyme: Mechanisms and a Brief Charge-spin Model

Although developing artificial enzymes has made great progress, there is still a gap between artificial enzymes and natural enzymes in catalytic performance. Designing and constructing efficient artificial biocatalysts is extremely desirable because of their high stability, low cost and easy storage. Here, we report a synthesized amino-functionalized graphene quantum dots-based manganese single atom catalyst (SAC) Mn-N4, which exhibits POD-, CAT, SOD-like activities, especially the superior SOD-like activity. Recent studies have reported Mn-based SAzymes, however, the multi-enzyme mimicking catalytic mechanisms for Mn-N4 are not comprehensive and in-depth enough. Therefore, we combine density functional theory (DFT) calculations and machine learning (ML) to validate the performance of the multi-enzyme mimicking activities. The DFT simulations show that Mn-N4 owns a highly effective SOD in the "one-side adsorption" with a very low energy barrier of 0.077 eV, which can be attributed to variation of the preferred spin states of Mn-O2.- system and its "spin flip-collection lock" in the SOD-like catalytic procedure. Furthermore, spin related charge distributions on Mn-N4 configurations by machine learning (ML) analysis suggest that the pattern of spin and natural charge/valence electron distribution will exhibit similarity in the structures of multiple intermediate steps of multi-enzyme mimicking activities. This work not only puts forward the catalytic mechanisms of Mn-N4 SAzymes, but also provides essential guidance for future design of highly performance artificial enzymes.

cond-mat.mes-hall↗

arXiv2023

On the $A_α$-spectral radius of graphs without linear forests

Let $A(G)$ and $D(G)$ be the adjacency and degree matrices of a simple graph $G$ on $n$ vertices, respectively. The \emph{$A_α$-spectral radius} of $G$ is the largest eigenvalue of $A_α(G)=αD(G)+(1-α)A(G)$ for a real number $α\in[0,1]$. In this paper, for $α\in (0,1)$, we obtain a sharp upper bound for the $A_α$-spectral radius of graphs on $n$ vertices without a subgraph isomorphic to a liner forest for $n$ large enough and characterize all graphs which attain the upper bound. As a result, we completely obtain the maximum signless Laplacian spectral radius of graphs on $n$ vertices without a subgraph isomorphic to a liner forest for $n$ large enough.

math.CO↗

arXiv2022

On weighted spectral radius of unraveled balls and normalized Laplacian eigenvalues

For a graph $G$, the unraveled ball of radius $r$ centered at a vertex $v$ is the ball of radius $r$ centered at $v$ in the universal cover of $G$. We obtain a lower bound on the weighted spectral radius of unraveled balls of fixed radius in a graph with positive weights on edges, which is used to present an upper bound on the $s$-th (where $s\ge 2$) smallest normalized Laplacian eigenvalue of irregular graphs under minor assumptions. Moreover, when $s=2$, the result may be regarded as an Alon--Boppana type bound for a class of irregular graphs.

math.CO↗

arXiv2022

The signless Laplacian spectral radius of graphs without trees

Let $Q(G)=D(G)+A(G)$ be the signless Laplacian matrix of a simple graph of order $n$, where $D(G)$ and $A(G)$ are the degree diagonal matrix and the adjacency matrix of $G$, respectively. In this paper, we present a sharp upper bound for the signless spectral radius of $G$ without any tree and characterize all extremal graphs which attain the upper bound, which may be regarded as a spectral extremal version for the famous Erdős-Sós conjecture.

math.CO↗