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Linyuan Lu

Publications and source records attributed to Linyuan Lu.

At least 19 recordsLinked to original sources
arXiv2026

On the classification of regular graphs with positive Lin-Lu-Yau curvature

We prove several new structural and classification results about $d$-regular graphs with positive Lin--Lu--Yau (LLY) curvature. We show that any positively curved $d$-regular graph has diameter at most $2d-2$, which improves the previously best known diameter bound obtained from the Bonnet-Myers-type theorem for positively curved graphs. We further show that every positively curved $d$-regular graph with $d\ge 3$ is $3$-connected. We classify all positively curved $3$-regular graphs, as well as all positively curved regular planar graphs.

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arXiv2026

Maximum spread of $K_{s,t}$-minor-free graphs II: the non-admissible cases

We have previously determined the maximum-spread $K_{s, t}$-minor-free graph(s) on $n$ vertices when $n$ is sufficiently large, $2\le s\le t$, and $s=2$ or $t\ge \frac{3}{2}(s-3) + \frac{4}{s-1}$. In this sequel paper, we completely determine the maximum-spread $K_{s, t}$-minor-free graphs on $n$ vertices for $n$ sufficiently large and $2\le s\le t$. In all of the remaining cases, the extremal graph is unique and is of the form $(K_r \vee (s-1-r)K_1) \vee (\ell_r K_t \cup (n-s+1-t\ell_r)K_1)$, where $r$ is an integer determined by $s$ and $t$ and $\ell_r$ is an integer determined by $n, s, t,$ and $r$.

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arXiv2026

Generalized Nordhaus--Gaddum Inequalities for Eigenvalues

For a graph $G$, let $ \lambda_1(G)\ge \lambda_2(G)\ge \cdots \ge \lambda_n(G)$ denote the adjacency eigenvalues of $G$. We investigate the asymptotic maximum of \[ \lambda_i(G)+\lambda_j(\overline G) \] for fixed $i$ and $j$. We prove general bounds on $\lambda_i(G) + \lambda_{j}(\overline{G})$ for all pairs $(i, j)$ and also give general bounds on the related problem of minimizing $\lambda_{n-i+1}(G) + \lambda_{n-j+1}(\overline{G})$ for fixed $i$ and $j$. We prove that for all looped graphs $G$ on $n$ vertices, \[\lambda_1(G) + \lambda_2(\overline{G}) \le \frac87 n. \] Our method also gives a new short proof of the Nordhaus-Gaddum result for the spectral radius proved by Terpai that $\lambda_1(G) + \lambda_1(\overline{G}) \le \frac43n - 1$. We also show the close relation of these Nordhaus-Gaddum type problems to recent work on the maximum spectral gaps of graphs by Brooks, Linz and Lu.

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arXiv2026

Tur\'an-Type Bounds for Graphs Containing Large $F$-Sparse Sets

We study Tur\'an-type extremal problems for graphs containing a large $F$-sparse vertex set, meaning a vertex set whose induced subgraph contains few copies of $F$. For integers $r>s\ge 1$, we prove that if a $K_{r+1}$-free graph $G$ on $n$ vertices contains a set $M$ of size $m\ge \lceil sn/r\rceil$ such that $G[M]$ is $K_{s+1}$-free, then \[ e(G)\le m(n-m)+t_s(m)+t_{r-s}(n-m). \] We characterize the equality cases as the complete $r$-partite graphs whose vertex classes split into two balanced groups of total sizes $m$ and $n-m$, consisting of $s$ and $r-s$ classes, respectively. We also prove a color-critical extension for forbidden graphs that embed into a join of two edge-critical graphs, together with an asymptotic extension for general $H$-free graphs in which the prescribed large vertex set spans few copies of a fixed graph $F$ with $\chi(F)<\chi(H)$.

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arXiv2025

An Ore-type theorem for $[3]$-graphs

Ore's Theorem states that if $G$ is an $n$-vertex graph and every pair of non-adjacent vertices has degree sum at least $n$, then $G$ is Hamiltonian. A $[3]$-graph is a hypergraph in which every edge contains at most $3$ vertices. In this paper, we prove an Ore-type result on the existence of Hamiltonian Berge cycles in $[3]$-graph $\cH$, based on the degree sum of every pair of non-adjacent vertices in the $2$-shadow graph $\partial \cH$ of $\cH$. Namely, we prove that there exists a constant $d_0$ such that for all $n \geq 6$, if a $[3]$-graph $\cH$ on $n$ vertices satisfies that every pair $u,v \in V(\cH)$ of non-adjacent vertices has degree sum $d_{\partial \cH}(u) + d_{\partial \cH}(v) \geq n+d_0$, then $\cH$ contains a Hamiltonian Berge cycle. Moreover, we conjecture that $d_0=1$ suffices.

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arXiv2024

Ricci Curvature Formula: Applications to Bonnet-Myers Sharp Irregular Graphs

In this paper, we establish a simple formula for computing the Lin-Lu-Yau Ricci curvature on graphs. For any edge $xy$ in a simple locally finite graph $G$, the curvature $\kappa(x,y)$ can be expressed as a cost function of an optimal bijection between two blow-up sets of the neighbors of $x$ and $y$. Utilizing this approach, we derive several results including a structural theorem for the Bonnet-Myers sharp irregular graphs of diameter $3$ and a theorem on $C_3$-free Bonnet-Myers sharp graphs.

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arXiv2024

Maximum spectral gaps of graphs

The spread of a graph $G$ is the difference $\lambda_1 - \lambda_n$ between the largest and smallest eigenvalues of its adjacency matrix. Breen, Riasanovsky, Tait and Urschel recently determined the graph on $n$ vertices with maximum spread for sufficiently large $n$. In this paper, we study a related question of maximizing the difference $\lambda_{i+1} - \lambda_{n-j}$ for a given pair $(i, j)$ over all graphs on $n$ vertices. We give upper bounds for all pairs $(i, j)$, exhibit an infinite family of pairs where the bound is tight, and show that for the pair $(1, 0)$ the extremal example is unique. These results contribute to a line of inquiry pioneered by Nikiforov aiming to maximize different linear combinations of eigenvalues over all graphs on $n$ vertices.

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arXiv2024

Maximum spread of $K_{s,t}$-minor-free graphs

The spread of a graph $G$ is the difference between the largest and smallest eigenvalue of the adjacency matrix of $G$. In this paper, we consider the family of graphs which contain no $K_{s,t}$-minor. We show that for any $t\geq s \geq 2$ and sufficiently large $n$, there is an integer $\xi_{t}$ such that the extremal $n$-vertex $K_{s,t}$-minor-free graph attaining the maximum spread is the graph obtained by joining a graph $L$ on $(s-1)$ vertices to the disjoint union of $\lfloor \frac{2n+\xi_{t}}{3t}\rfloor$ copies of $K_t$ and $n-s+1 - t\lfloor \frac{2n+\xi_t}{3t}\rfloor$ isolated vertices. Furthermore, we give an explicit formula for $\xi_{t}$ and an explicit description for the graph $L$ for $t \geq \frac32(s-3) +\frac{4}{s-1}$.

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arXiv2023

On the maximum second eigenvalue of outerplanar graphs

For a fixed positive integer $k$ and a graph $G$, let $\lambda_k(G)$ denote the $k$-th largest eigenvalue of the adjacency matrix of $G$. In 2017, Tait and Tobin proved that the maximum $\lambda_1(G)$ among all outerplanar graphs on $n$ vertices is achieved by the fan graph $K_1\vee P_{n-1}$. In this paper, we consider a similar problem of determining the maximum $\lambda_2$ among all connected outerplanar graphs on $n$ vertices. For $n$ even and sufficiently large, we prove that the maximum $\lambda_2$ is uniquely achieved by the graph $(K_1\vee P_{n/2-1})\!\!-\!\!(K_1\vee P_{n/2-1})$, which is obtained by connecting two disjoint copies of $(K_1\vee P_{n/2-1})$ through a new edge joining their smallest degree vertices. When $n$ is odd and sufficiently large, the extremal graphs are not unique. The extremal graphs are those graphs $G$ that contain a cut vertex $u$ such that $G\setminus \{u\}$ is isomorphic to $2(K_1\vee P_{n/2-1})$. We also determine the maximum $\lambda_2$ among all 2-connected outerplanar graphs and asymptotically determine the maximum of $\lambda_k(G)$ among all connected outerplanar graphs for any fixed $k$.

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arXiv2023

Maximum spread of $K_{2,t}$-minor-free graphs

The spread of a graph $G$ is the difference between the largest and smallest eigenvalues of the adjacency matrix of $G$. In this paper, we consider the family of graphs which contain no $K_{2,t}$-minor. We show that for any $t\geq 2$, there is an integer $ξ_t$ such that the maximum spread of an $n$-vertex $K_{2,t}$-minor-free graph is achieved by the graph obtained by joining a vertex to the disjoint union of $\lfloor \frac{2n+ξ_t}{3t}\rfloor$ copies of $K_t$ and $n-1 - t\lfloor \frac{2n+ξ_t}{3t}\rfloor$ isolated vertices. The extremal graph is unique, except when $t\equiv 4 \mod 12$ and $\frac{2n+ ξ_t} {3t}$ is an integer, in which case the other extremal graph is the graph obtained by joining a vertex to the disjoint union of $\lfloor \frac{2n+ξ_t}{3t}\rfloor-1$ copies of $K_t$ and $n-1-t(\lfloor \frac{2n+ξ_t}{3t}\rfloor-1)$ isolated vertices. Furthermore, we give an explicit formula for $ξ_t$.

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arXiv2023

Optimal Ricci curvature Markov chain Monte Carlo methods on finite states

We construct a new Markov chain Monte Carlo method on finite states with optimal choices of acceptance-rejection ratio functions. We prove that the constructed continuous time Markov jumping process has a global in-time convergence rate in $L^1$ distance. The convergence rate is no less than one-half and is independent of the target distribution. For example, our method recovers the Metropolis-Hastings algorithm on a two-point state. And it forms a new algorithm for sampling general target distributions. Numerical examples are presented to demonstrate the effectiveness of the proposed algorithm.

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arXiv2022

On the maximum spread of planar and outerplanar graphs

The spread of a graph $G$ is the difference between the largest and smallest eigenvalue of the adjacency matrix of $G$. Gotshall, O'Brien and Tait conjectured that for sufficiently large $n$, the $n$-vertex outerplanar graph with maximum spread is the graph obtained by joining a vertex to a path on $n-1$ vertices. In this paper, we disprove this conjecture by showing that the extremal graph is the graph obtained by joining a vertex to a path on $\lceil (2n-1)/3\rceil$ vertices and $\lfloor(n-2)/3\rfloor$ isolated vertices. For planar graphs, we show that the extremal $n$-vertex planar graph attaining the maximum spread is the graph obtained by joining two nonadjacent vertices to a path on $\lceil(2n-2)/3\rceil$ vertices and $\lfloor(n-4)/3\rfloor$ isolated vertices.

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arXiv2022

Mean field information Hessian matrices on graphs

We derive mean-field information Hessian matrices on finite graphs. The "information" refers to entropy functions on the probability simplex. And the "mean-field" means nonlinear weight functions of probabilities supported on graphs. These two concepts define a mean-field optimal transport type metric. In this metric space, we first derive Hessian matrices of energies on graphs, including linear, interaction energies, entropies. We name their smallest eigenvalues as mean-field Ricci curvature bounds on graphs. We next provide examples on two-point spaces and graph products. We last present several applications of the proposed matrices. E.g., we prove discrete Costa's entropy power inequalities on a two-point space.

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arXiv2021

Anti-Ramsey number of disjoint rainbow bases in all matroids

Consider a matroid $M=(E,\mathcal{I})$ with its elements of the ground set $E$ colored. A rainbow basis is a maximum independent set in which each element receives a different color. The rank of a subset $S$ of $E$, denoted by $r_M(S)$, is the maximum size of an independent set in $S$. A flat $F$ is a maximal set in $M$ with a fixed rank. The anti-Ramsey number of $t$ pairwise disjoint rainbow bases in $M$, denoted by $ar(M,t)$, is defined as the maximum number of colors $m$ such that there exists an $m$ coloring of the ground set $E$ of $M$ which contains no $t$ pairwise disjoint rainbow bases. We determine $ar(M,t)$ for all matroids of rank at least 2: $ar(M,t)=|E|$ if there exists a flat $F_0$ with $|E|-|F_0|<t(r_M(E)-r_M(F_0))$; and $ar(M,t)=\max_{F\colon r_M(F)\leq r_M(E)-2} \{|F|+t(r_M(E)-r_M(F)-1)\}$ otherwise. This generalizes Lu-Meier-Wang's previous result on the anti-Ramsey number of edge-disjoint rainbow spanning trees in any multigraph $G$.

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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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