arXiv subjects

Xiaona Fang

Publications and source records attributed to Xiaona Fang.

12 recordsLinked to original sources

arXiv2026

Any $k$-graph with zero $\ell$-degree Turán density is layered

The codegree Turán density $π_{\mathrm{co}}(F)$ is the supremum over all $γ\in [0,1)$ such that, for arbitrarily large $n$, there exists an $n$-vertex $F$-free $k$-graph $H$ whose every $(k-1)$-subset of vertices lies in at least $γn$ edges. Ding, Lamaison, Liu, Wang, and Yang (JLMS, 2025) studied the problem of what 3-graphs $F$ satisfy $π_{\mathrm{co}}(F) = 0$. They introduced layered $3$-graphs and conjectured that a $3$-graph has zero codegree Turán density if and only if it is layered and has zero uniform Turán density. For $k\ge 3$, a $k$-graph is called layered if its vertices can be labelled so that every edge has a unique maximum label and two edges with the same maximum label have the same label multiset. In this paper, we show that every non-layered $k$-graph $F$ on $m$ vertices satisfies \[ π_{\mathrm{co}}(F)\ge q_{k,m}^{-q_{k,m}}>0, \quad \text{where}\quad q_{k,m}=\frac{(k-1)^{m+1}-1}{k-2}, \] which implies any $k$-graph with zero $\ell$-degree Turán density is layered, and the case $k=3$ confirms the conjecture of Ding, Lamaison, Liu, Wang, and Yang.

math.CO↗

arXiv2026

On the codegree Turán density of projective geometries

Let $F$ be a $k$-uniform hypergraph, abbreviated as $k$-graph. The codegree Turán density $γ(F)$ is the supremum over all $γ\in [0,1)$ such that, for arbitrarily large $n$, there exists an $n$-vertex $F$-free $k$-graph $H$ whose every $(k-1)$-subset of vertices lies in at least $γn$ edges. Let $PG_m(q)$ be the projective geometry of dimension $m$ over finite field $\mathbb{F}_q$. In this paper, we prove that $γ(PG_m(q)) \ge \frac{1}{p}> 0$ for all $m$ and $q$, where $p$ is the smallest prime divisor of $q+1$. This resolves an open problem proposed by Keevash and Zhao (JCT-B, 2007). Moreover, we determine the exact codegree Turán density of $PG_4(q)$ when $q$ is an odd prime power.

math.CO↗

arXiv2026

On linear $k$-graphs with codegree Turán density arbitrarily close to zero

Let $F$ be a $k$-uniform hypergraph, abbreviated as $k$-graph. The codegree Turán density $π_{\mathrm{co}}(F)$ is the supremum over all $γ\in [0,1)$ such that, for arbitrarily large $n$, there exists an $n$-vertex $F$-free $k$-graph $H$ whose every $(k-1)$-subset of vertices lies in at least $γn$ edges. In this paper, we prove that there is a linear $k$-graph $F$ with $0<π_{co}(F) < \varepsilon$ for any $\varepsilon>0$. The special case $k=3$ solve a question proposed by Ding, Lamaison, Liu, Wang and Yang (JLMS, 2025). The main method combines an affine-plane-type incidence structure over a finite field and elementary number-theoretic arguments.

math.CO↗

arXiv2025

Holistic structure of neural pathways underlies brain perceptual rivalry: Physical mechanism of auditory stream segregation

Brain perceptual rivalry, exemplified by auditory stream segregation of competing tones (A_, B__, ABA_), serves as a core mechanism of brain perception formation. While increasingly recognized as determining by neural connections rather than specific neural groups, the mechanism of brain perception remains uncertain. We demonstrate that auditory stream segregation arises from the topological structure of holistic neural pathways. By constructing a holistic pathway model using existing neurophysiological data, combining nonlinear neural dynamics and nonequilibrium physics, we uncover the biophysical mechanism of perceptual phase transitions from integrated (ABA_) to segregated streams (A_ or B_), as well as the mechanism of temporal dynamics, perceptual switching path, and attention regulation underlying these transitions. Further, we demonstrate how our framework reveals energy consumption of the auditory system and combines it with neuroelectrophysiology. Two psycho-acoustic experiments validate our predictions of perception alternation and attention modulation. Our framework provides a transformative perspective on how brain networks generate complex perceptual experiences, emphasizing the significance of neural pathway structure in the process of brain function realization.

q-bio.NC↗

arXiv2024

The Turán number of path-star forests

The Turán number of a graph $H$, denoted by $ex(n,H)$, is the maximum number of edges in any graph on $n$ vertices containing no $H$ as a subgraph. A linear (star) forest is a forest consisting of paths (stars). A path-star forest $F$ is a forest consisting of paths and stars. In this paper, we determine $ex(n,F)$ for sufficiently large $n$ and characterize the corresponding extremal graphs, and our result generalizes previous known results on the Turán numbers of linear forests and star forests.

math.CO↗

arXiv2024

Generalized Turán problem for a path and a clique

Let $\mathcal{H}$ be a family of graphs. The generalized Turán number $ex(n, K_r, \mathcal{H})$ is the maximum number of copies of the clique $K_r$ in any $n$-vertex $\mathcal{H}$-free graph. In this paper, we determine the value of $ex(n, K_r, \{P_k, K_m \} )$ for sufficiently large $n$ with an exceptional case, and characterize all corresponding extremal graphs, which generalizes and strengthens the results of Katona and Xiao [EJC, 2024] on $ex(n, K_2, \{P_k, K_m \} )$. For the exceptional case, we obtain a tight upper bound for $ex(n, K_r, \{P_k, K_m \} )$ that confirms a conjecture on $ex(n, K_2, \{P_k, K_m \} )$ posed by Katona and Xiao.

math.CO↗

arXiv2022

Regular graphs with a complete bipartite graph as a star complement

Let $G$ be a graph of order $n$ and $μ$ be an adjacency eigenvalue of $G$ with multiplicity $k\geq 1$. A star complement $H$ for $μ$ in $G$ is an induced subgraph of $G$ of order $n-k$ with no eigenvalue $μ$, and the vertex subset $X=V(G-H)$ is called a star set for $μ$ in $G$. The study of star complements and star sets provides a strong link between graph structure and linear algebra. In this paper, we study the regular graphs with $K_{t,s}\ (s\geq t\geq 2)$ as a star complement for an eigenvalue $μ$, especially, characterize the case of $t=3$ completely, obtain some properties when $t=s$, and propose some problems for further study.

math.CO↗

arXiv2022

The maximum spectral radius of graphs of given size with forbidden subgraph

Let $G$ be a graph of size $m$ and $ρ(G)$ be the spectral radius of its adjacency matrix. A graph is said to be $F$-free if it does not contain a subgraph isomorphic to $F$. In this paper, we prove that if $G$ is a $K_{2,r+1}$-free non-star graph with $m\geq (4r+2)^2+1$, then $ρ(G)\leq ρ(S_m^1)$, with equality if and only if $G\cong S_m^1$. Recently, Li, Sun and Wei showed that for any $θ_{1,2,3}$-free graph of size $m\geq 8$, $ρ(G)\leq \frac{1+\sqrt{4m-3}}{2}$, with equality if and only if $G\cong S_{\frac{m+3}{2},2}$. However, this bound is not attainable when $m$ is even. We proved that if $G$ is $θ_{1,2,3}$-free and $G\ncong S_{\frac{m+3}{2},2}$ with $m\geq 22$, then $ρ(G)\leq ρ(F_{m,1})$ if $m$ is even, with equality if and only if $G\cong F_{m,1}$, and $ρ(G)\leq ρ(F_{m,2})$ if $m$ is odd, with equality if and only if $G\cong F_{m,2}$.

math.CO↗

arXiv2021

Spectral properties of $p$-Sombor matrices and beyond

Let $G=(V(G),E(G))$ be a simple graph with vertex set $V(G)=\{v_{1},v_{2},\cdots, v_{n}\}$ and edge set $E(G)$. The $p$-Sombor matrix $\mathcal{S}_{p}(G)$ of $G$ is the square matrix of order $n$ whose $(i,j)$-entry is equal to $((d_{i})^{p}+(d_{j})^{p})^{\frac{1}{p}}$ if $v_{i}\sim v_{j}$, and 0 otherwise, where $d_{i}$ denotes the degree of vertex $v_{i}$ in $G$. In this paper, we study the relationship between $p$-Sombor index $SO_{p}(G)$ and $p$-Sombor matrix $\mathcal{S}_{p}(G)$ by the $k$-th spectral moment $N_{k}$ and the spectral radius of $\mathcal{S}_{p}(G)$. Then we obtain some bounds of $p$-Sombor Laplacian eigenvalues, $p$-Sombor spectral radius, $p$-Sombor spectral spread, $p$-Sombor energy and $p$-Sombor Estrada index. We also investigate the Nordhaus-Gaddum-type results for $p$-Sombor spectral radius and energy. At last, we give the regression model for boiling point and some other invariants.

math.CO↗

arXiv2021

The expected values of Sombor indices in random hexagonal chains, phenylene chains and Sombor indices of some chemical graphs

Hexagonal chains are a special class of catacondensed benzenoid system and phenylene chains are a class of polycyclic aromatic compounds. Recently, A family of Sombor indices was introduced by Gutman in the chemical graph theory. It had been examined that these indices may be successfully applied on modeling thermodynamic properties of compounds. In this paper, we study the expected values of the Sombor indices in random hexagonal chains, phenylene chains, and consider the Sombor indices of some chemical graphs such as graphene, coronoid systems and carbon nanocones.

math.CO↗

arXiv2020

Nonequilibrium Physics in Biology

Life is characterized by a myriad of complex dynamic processes allowing organisms to grow, reproduce, and evolve. Physical approaches for describing systems out of thermodynamic equilibrium have been increasingly applied to living systems, which often exhibit phenomena unknown from those traditionally studied in physics. Spectacular advances in experimentation during the last decade or two, for example, in microscopy, single cell dynamics, in the reconstruction of sub- and multicellular systems outside of living organisms, or in high throughput data acquisition have yielded an unprecedented wealth of data about cell dynamics, genetic regulation, and organismal development. These data have motivated the development and refinement of concepts and tools to dissect the physical mechanisms underlying biological processes. Notably, the landscape and flux theory as well as active hydrodynamic gel theory have proven very useful in this endeavour. Together with concepts and tools developed in other areas of nonequilibrium physics, significant progresses have been made in unraveling the principles underlying efficient energy transport in photosynthesis, cellular regulatory networks, cellular movements and organization, embryonic development and cancer, neural network dynamics, population dynamics and ecology, as well as ageing, immune responses and evolution. Here, we review recent advances in nonequilibrium physics and survey their application to biological systems. We expect many of these results to be important cornerstones as the field continues to build our understanding of life.

physics.bio-ph↗

arXiv2017

The emergence of the two cell fates and their associated switching for a negative auto-regulating gene

Decisions in the cell that lead to its ultimate fate are important for cellular functions such as proliferation, growth, differentiation, development and death. Understanding this decision process is imperative for advancements in the treatment of diseases such as cancer. It is clear that underlying gene regulatory networks and surrounding environments of the cells are crucial for function. The self-repressor is a very abundant gene regulatory motif, and is often believed to have only one cell fate. In this study, we elucidate the effects of microenvironments mimicking the epigenetic effects on cell fates through the introduction of inducers capable of binding to a self-repressing gene product (protein), thus regulating the associated gene. This alters the effective regulatory binding speed of the self-repressor regulatory protein to its destination DNA without changing the gene itself. The steady state observations and real time monitoring of the self-repressor expression dynamics reveal the emergence of the two cell fates, The simulations are consistent with the experimental findings. We provide physical and quantitative explanations for the origin of the two phenotypic cell fates. We find that two cell fates, rather than a single fate, and their associated switching dynamics emerge from a change in effective gene regulation strengths. The switching time scale is quantified. Our results reveal a new mechanism for the emergence of multiple cell fates. This provides an origin for the heterogeneity often observed among cell states, while illustrating the influence of microenvironments on cell fates and their decision-making processes without genetic changes

q-bio.MN↗