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Linyuan Lu

Publications and source records attributed to Linyuan Lu.

At least 37 records · Page 2Linked to original sources
arXiv2021

Maximum spectral radius of outerplanar 3-uniform hypergraphs

In this paper, we study the maximum spectral radius of outerplanar $3$-uniform hypergraphs. Given a hypergraph $\mathcal{H}$, the shadow of $\mathcal{H}$ is a graph $G$ with $V(G)= V(\mathcal{H})$ and $E(G) = \{uv: uv \in h \textrm{ for some } h\in E(\mathcal{H})\}$. A graph is \textit{outerplanar} if it can be embedded in the plane such that all its vertices lie on the outer face. A $3$-uniform hypergraph $\mathcal{H}$ is called \textit{outerplanar} if its shadow has an outerplanar embedding such that every hyperedge of $\mathcal{H}$ is the vertex set of an interior triangular face of the shadow. Cvetković and Rowlinson conjectured in 1990 that among all outerplanar graphs on $n$ vertices, the graph $K_1+ P_{n-1}$ attains the maximum spectral radius. We show a hypergraph analogue of the Cvetković-Rowlinson conjecture. In particular, we show that for sufficiently large $n$, the $n$-vertex outerplanar $3$-uniform hypergraph of maximum spectral radius is the unique $3$-uniform hypergraph whose shadow is $K_1 + P_{n-1}$.

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arXiv2021

Ricci-flat graphs with maximum degree at most 4

A graph is called Ricci-flat if its Ricci curvatures vanish on all edges, here the definition of Ricci curvature on graphs was given by Lin-Lu-Yau. The authors in arXiv:1301.0102 and arXiv:1802.02982 obtained a complete characterization for all Ricci-flat graphs with girth at least five. In this paper, we completely determined all Ricci-flat graphs with maximum degree at most 4.

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arXiv2021

Anti-Ramsey number of edge-disjoint rainbow spanning trees in all graphs

An edge-colored graph $G$ is called \textit{rainbow} if every edge of $G$ receives a different color. Given any host graph $G$, the \textit{anti-Ramsey} number of $t$ edge-disjoint rainbow spanning trees in $G$, denoted by $r(G,t)$, is defined as the maximum number of colors in an edge-coloring of $G$ containing no $t$ edge-disjoint rainbow spanning trees. For any vertex partition $P$, let $E(P,G)$ be the set of non-crossing edges in $G$ with respect to $P$. In this paper, we determine $r(G,t)$ for all host graphs $G$: $r(G,t)=|E(G)|$ if there exists a partition $P_0$ with $|E(G)|-|E(P_0,G)|<t(|P_0|-1)$; and $r(G,t)=\max_{P\colon |P|\geq 3} \{|E(P,G)|+t(|P|-2)\}$ otherwise. As a corollary, we determine $r(K_{p,q},t)$ for all values of $p,q, t$, improving a result of Jia, Lu and Zhang.

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arXiv2020

Turán Density of $2$-edge-colored Bipartite Graphs with Application on $\{2, 3\}$-Hypergraphs

We consider the Turán problems of $2$-edge-colored graphs. A $2$-edge-colored graph $H=(V, E_r, E_b)$ is a triple consisting of the vertex set $V$, the set of red edges $E_r$ and the set of blue edges $E_b$ with $E_r$ and $E_b$ do not have to be disjoint. The Turán density $π(H)$ of $H$ is defined to be $\lim_{n\to\infty} \max_{G_n}h_n(G_n)$, where $G_n$ is chosen among all possible $2$-edge-colored graphs on $n$ vertices containing no $H$ as a sub-graph and $h_n(G_n)=(|E_r(G)|+|E_b(G)|)/{n\choose 2}$ is the formula to measure the edge density of $G_n$. We will determine the Turán densities of all $2$-edge-colored bipartite graphs. We also give an important application of our study on the Turán problems of $\{2, 3\}$-hypergraphs.

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arXiv2020

On the Sum of Ricci-Curvatures for Weighted Graphs

In this paper, we generalize Lin-Lu-Yau's Ricci curvature to weighted graphs and give a simple limit-free definition. We prove two extremal results on the sum of Ricci curvatures for weighted graph. A weighted graph $G=(V,E,d)$ is an undirected graph $G=(V,E)$ associated with a distance function $d\colon E\to [0,\infty)$. By redefining the weights if possible, without loss of generality, we assume that the shortest weighted distance between $u$ and $v$ is exactly $d(u,v)$ for any edge $uv$. Now consider a random walk whose transitive probability from an vertex $u$ to its neighbor $v$ (a jump move along the edge $uv$) is proportional to $w_{uv}:=F(d(u,v))/d(u,v)$ for some given function $F(\bullet)$. We first generalize Lin-Lu-Yau's Ricci curvature definition to this weighted graph and give a simple limit-free representation of $κ(x, y)$ using a so called $\ast$-coupling functions. The total curvature $K(G)$ is defined to be the sum of Ricci curvatures over all edges of $G$. We proved the following theorems: if $F(\bullet)$ is a decreasing function, then $K(G)\geq 2|V| -2|E|$; if $F(\bullet)$ is an increasing function, then $K(G)\leq 2|V| -2|E|$. Both equalities hold if and only if $d$ is a constant function plus the girth is at least $6$. In particular, these imply a Gauss-Bonnet theorem for (unweighted) graphs with girth at least $6$, where the graph Ricci curvature is defined geometrically in terms of optimal transport.

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arXiv2020

On the size of planar graphs with positive Lin-Lu-Yau Ricci curvature

We show that if a planar graph $G$ with minimum degree at least $3$ has positive Lin-Lu-Yau Ricci curvature on every edge, then $Δ(G)\leq 17$, which then implies that $G$ is finite. This is an analogue of a result of DeVos and Mohar [{\em Trans. Amer. Math. Soc., 2007}] on the size of planar graphs with positive combinatorial curvature.

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arXiv2020

On Hamiltonian Berge cycles in $3$-uniform hypergraphs

Given a set $R$, a hypergraph is $R$-uniform if the size of every hyperedge belongs to $R$. A hypergraph $\mathcal{H}$ is called \textit{covering} if every vertex pair is contained in some hyperedge in $\mathcal{H}$. In this note, we show that every covering $[3]$-uniform hypergraph on $n\geq 6$ vertices contains a Berge cycle $C_s$ for any $3\leq s\leq n$. As an application, we determine the maximum Lagrangian of $k$-uniform Berge-$C_{t}$-free hypergraphs and Berge-$P_{t}$-free hypergraphs.

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arXiv2019

Anti-Ramsey number of edge-disjoint rainbow spanning trees

An edge-colored graph $G$ is called rainbow if every edge of $G$ receives a different color. The anti-Ramsey number of $t$ edge-disjoint rainbow spanning trees, denoted by $r(n,t)$, is defined as the maximum number of colors in an edge-coloring of $K_n$ containing no $t$ edge-disjoint rainbow spanning trees. Jahanbekam and West [J. Graph Theory, 2014] conjectured that for any fixed $t$, $r(n,t)=\binom{n-2}{2}+t$ whenever $n\geq 2t+2 \geq 6$. In this paper, we prove this conjecture. We also determine $r(n,t)$ when $n = 2t+1$. Together with previous results, this gives the anti-Ramsey number of $t$ edge-disjoint rainbow spanning trees for all values of $n$ and $t$.

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arXiv2019

Poset Ramsey Numbers for Boolean Lattices

A subposet $Q'$ of a poset $Q$ is a \textit{copy of a poset} $P$ if there is a bijection $f$ between elements of $P$ and $Q'$ such that $x \le y$ in $P$ iff $f(x) \le f(y)$ in $Q'$. For posets $P, P'$, let the \textit{poset Ramsey number} $R(P,P')$ be the smallest $N$ such that no matter how the elements of the Boolean lattice $Q_N$ are colored red and blue, there is a copy of $P$ with all red elements or a copy of $P'$ with all blue elements. Axenovich and Walzer introduced this concept in \textit{Order} (2017), where they proved $R(Q_2, Q_n) \le 2n + 2$ and $R(Q_n, Q_m) \le mn + n + m$, where $Q_n$ is the Boolean lattice of dimension $n$. They later proved $2n \le R(Q_n, Q_n) \le n^2 + 2n$. Walzer later proved $R(Q_n, Q_n) \le n^2 + 1$. We provide some improved bounds for $R(Q_n, Q_m)$ for various $n,m \in \mathbb{N}$. In particular, we prove that $R(Q_n, Q_n) \le n^2 - n + 2$, $R(Q_2, Q_n) \le \frac{5}{3}n + 2$, and $R(Q_3, Q_n) \le \frac{37}{16}n + \frac{39}{16}$. We also prove that $R(Q_2,Q_3) = 5$, and $R(Q_m, Q_n) \le (m - 2 + \frac{9m - 9}{(2m - 3)(m + 1)})n + m + 3$ for all $n \ge m \ge 4$.

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arXiv2019

Concentration inequalities in spaces of random configurations with positive Ricci curvatures

In this paper, we prove an Azuma-Hoeffding-type inequality in several classical models of random configurations, including the Erdős-Rényi random graph models $G(n,p)$ and $G(n,M)$, the random $d$-out(in)-regular directed graphs, and the space of random permutations. The main idea is using Ollivier's work on the Ricci curvature of Markov chairs on metric spaces. Here we give a cleaner form of such concentration inequality in graphs. Namely, we show that for any Lipschitz function $f$ on any graph (equipped with an ergodic random walk and thus an invariant distribution $ν$) with Ricci curvature at least $κ>0$, we have \[ν\left( |f-E_νf| \geq t \right) \leq 2\exp\left( -\frac{t^2κ}{7} \right).\]

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arXiv2019

On the cover Turán number of Berge hypergraphs

For a fixed set of positive integers $R$, we say $\mathcal{H}$ is an $R$-uniform hypergraph, or $R$-graph, if the cardinality of each edge belongs to $R$. For a graph $G=(V,E)$, a hypergraph $\mathcal{H}$ is called a Berge-$G$, denoted by $BG$, if there exists a bijection $f: E(G) \to E(\mathcal{H})$ such that for every $e \in E(G)$, $e \subseteq f(e)$. In this paper, we define a variant of Turán number in hypergraphs, namely the cover Turán number, denoted as $\hat{ex}_R(n, G)$, as the maximum number of edges in the shadow graph of a Berge-$G$ free $R$-graph on $n$ vertices. We show a general upper bound on the cover Turán number of graphs and determine the cover Turán density of all graphs when the uniformity of the host hypergraph equals to $3$.

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arXiv2019

On the cover Ramsey number of Berge hypergraphs

For a fixed set of positive integers $R$, we say $\mathcal{H}$ is an $R$-uniform hypergraph, or $R$-graph, if the cardinality of each edge belongs to $R$. An $R$-graph $\mathcal{H}$ is \emph{covering} if every vertex pair of $\mathcal{H}$ is contained in some hyperedge. For a graph $G=(V,E)$, a hypergraph $\mathcal{H}$ is called a \textit{Berge}-$G$, denoted by $BG$, if there exists an injection $f: E(G) \to E(\mathcal{H})$ such that for every $e \in E(G)$, $e \subseteq f(e)$. In this note, we define a new type of Ramsey number, namely the \emph{cover Ramsey number}, denoted as $\hat{R}^R(BG_1, BG_2)$, as the smallest integer $n_0$ such that for every covering $R$-uniform hypergraph $\mathcal{H}$ on $n \geq n_0$ vertices and every $2$-edge-coloring (blue and red) of $\mathcal{H}$ , there is either a blue Berge-$G_1$ or a red Berge-$G_2$ subhypergraph. We show that for every $k\geq 2$, there exists some $c_k$ such that for any finite graphs $G_1$ and $G_2$, $R(G_1, G_2) \leq \hat{R}^{[k]}(BG_1, BG_2) \leq c_k \cdot R(G_1, G_2)^3$. Moreover, we show that for each positive integer $d$ and $k$, there exists a constant $c = c(d,k)$ such that if $G$ is a graph on $n$ vertices with maximum degree at most $d$, then $\hat{R}^{[k]}(BG,BG) \leq cn$.

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arXiv2018

The extremal $p$-spectral radius of Berge-hypergraphs

Let $G$ be a graph. We say that a hypergraph $H$ is a Berge-$G$ if there is a bijection $ϕ: E(G)\to E(H)$ such that $e\subseteq ϕ(e)$ for all $e\in E(G)$. For any $r$-uniform hypergraph $H$ and a real number $p\geq 1$, the $p$-spectral radius $λ^{(p)}(H)$ of $H$ is defined as \[ λ^{(p)}(H):=\max_{{\bf x}\in\mathbb{R}^n,\,\|{\bf x}\|_p=1} r\sum_{\{i_1,i_2,\ldots,i_r\}\in E(H)} x_{i_1}x_{i_2}\cdots x_{i_r}. \] In this paper, we study the $p$-spectral radius of Berge-$G$ hypergraphs. We determine the $3$-uniform hypergraphs with maximum $p$-spectral radius for $p\geq 1$ among Berge-$G$ hypergraphs when $G$ is a path, a cycle or a star.

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arXiv2018

On a hypergraph probabilistic graphical model

We propose a directed acyclic hypergraph framework for a probabilistic graphical model that we call Bayesian hypergraphs. The space of directed acyclic hypergraphs is much larger than the space of chain graphs. Hence Bayesian hypergraphs can model much finer factorizations than Bayesian networks or LWF chain graphs and provide simpler and more computationally efficient procedures for factorizations and interventions. Bayesian hypergraphs also allow a modeler to represent causal patterns of interaction such as Noisy-OR graphically (without additional annotations). We introduce global, local and pairwise Markov properties of Bayesian hypergraphs and prove under which conditions they are equivalent. We define a projection operator, called shadow, that maps Bayesian hypergraphs to chain graphs, and show that the Markov properties of a Bayesian hypergraph are equivalent to those of its corresponding chain graph. We extend the causal interpretation of LWF chain graphs to Bayesian hypergraphs and provide corresponding formulas and a graphical criterion for intervention.

cs.DS↗
arXiv2018

High-order Phase Transition in Random Hypergrpahs

In this paper, we study the high-order phase transition in random $r$-uniform hypergraphs. For a positive integer $n$ and a real $p\in [0,1]$, let $H:=H^r(n,p)$ be the random $r$-uniform hypergraph with vertex set $[n]$, where each $r$-set is selected as an edge with probability $p$ independently randomly. For $1\leq s \leq r-1$ and two $s$-sets $S$ and $S'$, we say $S$ is connected to $S'$ if there is a sequence of alternating $s$-sets and edges $S_0,F_1,S_1,F_2, \ldots, F_k, S_k$ such that $S_0,S_1,\ldots, S_k$ are $s$-sets, $S_0=S$, $S_k=S'$, $F_1,F_2,\ldots, F_k$ are edges of $H$, and $S_{i-1}\cup S_i\subseteq F_i$ for each $1\leq i\leq k$. This is an equivalence relation over the family of all $s$-sets ${[n]\choose s}$ and results in a partition: ${V\choose s}=\cup_i C_i$. Each $C_i$ is called an { $s$-th-order} connected component and a component $C_i$ is {\em giant} if $|C_i|=Θ(n^s)$. We prove that the sharp threshold of the existence of the $s$-th-order giant connected components in $H^r(n,p)$ is $\frac{1}{\big({r\choose s}-1\big){n\choose r-s}}$. Let $c={n\choose r-s}p$. If $c$ is a constant and $c<\tfrac{1}{\binom{r}{s}-1}$, then with high probability, all $s$-th-order connected components have size $O(\ln n)$. If $c$ is a constant and $c > \tfrac{1}{\binom{r}{s}-1}$, then with high probability, $H^r(n,p)$ has a unique giant connected $s$-th-order component and its size is $(z+o(1)){n\choose s}$, where $$z=1-\sum_{j=0}^\infty \frac{\left({r\choose s}j -j+1 \right)^{j-1}}{j!}c^je^{-c\left({r\choose s}j -j+1\right)}.$$

math.CO↗
arXiv2018

On Hypergraph Lagrangians and Frankl-Füredi's Conjecture

Frankl and Füredi conjectured in 1989 that the maximum Lagrangian, denoted by $λ_r(m)$, among all $r$-uniform hypergraphs of fixed size $m$ is achieved by the minimum hypergraph $C_{r,m}$ under the colexicographic order. We say $m$ in {\em principal domain} if there exists an integer $t$ such that ${t-1\choose r}\leq m\leq {t\choose r}-{t-2\choose r-2}$. If $m$ is in the principal domain, then Frankl-Füredi's conjecture has a very simple expression: $$λ_r(m)=\frac{1}{(t-1)^r}{t-1\choose r}.$$ Many previous results are focusing on $r=3$. For $r\geq 4$, Tyomkyn in 2017 proved that Frankl-Füredi's conjecture holds whenever ${t-1\choose r} \leq m \leq {t\choose r} -{t-2\choose r-2}- δ_rt^{r-2}$ for a constant $δ_r>0$. In this paper, we improve Tyomkyn's result by showing Frankl-Füredi's conjecture holds whenever ${t-1\choose r} \leq m \leq {t\choose r} -{t-2\choose r-2}- δ_r't^{r-\frac{7}{3}}$ for a constant $δ_r'>0$.

math.CO↗
arXiv2018

On Lagrangians of $3$-uniform hypergraphs

Frankl and Füredi conjectured in 1989 that the maximum Lagrangian of all $r$-uniform hypergraphs of fixed size $m$ is realized by the minimum hypergraph $C_{r,m}$ under the colexicographic order. In this paper, we prove a weaker version of the Frankl and Füredi's conjecture at $r=3$: there exists an absolute constant $c>0$ such that for any $3$-uniform hypergraph $H$ with $m$ edges, the Lagrangian of $H$ satisfies $λ(H)\leq λ(C_{3,m+cm^{2/9}})$. In particular, this result implies that the Frankl and Füredi's conjecture holds for $r=3$ and $m\in [{t-1\choose 3}, {t\choose 3}-(t-2)-ct^{\frac{2}{3}}]$. It improves a recent result of Tyomkyn.

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