arXiv subjects

Ruiling Zheng

Publications and source records attributed to Ruiling Zheng.

10 recordsLinked to original sources

arXiv2026

A sharp Randi\'c bound for K\"onig--Egerv\'ary graphs and a conjecture of Aouchiche, Hansen, and Zheng

Let $\alpha'(G)$ be the matching number of a graph $G$, and let its Randi\'c index be $R(G)=\sum_{uv\in E(G)}(d(u)d(v))^{-1/2}$. In 2006, Aouchiche, Hansen, and Zheng conjectured that the maximum of $R(G)-\alpha'(G)$ over all $n$-vertex graphs is attained by the complete bipartite graph whose smaller part has $\lfloor\frac{n+4}{7}\rfloor$ vertices; the conjecture has remained open since then. In this paper, we prove that every $n$-vertex K\"onig--Egerv\'ary graph, and in particular every bipartite graph, satisfies \[ R(G)\le\sqrt{\alpha'(G)\left(n-\alpha'(G)\right)}, \] and we characterize the graphs attaining equality as the bipartite graphs all of whose components are semiregular with a common degree ratio. The K\"onig--Egerv\'ary hypothesis cannot be dropped, but the Berge--Tutte formula reduces the general case to it, and in this way we determine the maximum of $R(G)-\alpha'(G)$ for every $n\ge4$, together with all extremal graphs. The conjecture is therefore false, and it fails for infinitely many orders: the optimal part size is governed by the proportion $\frac{2-\sqrt2}{4}$ rather than by $\frac17$. The two proportions give asymptotic slopes differing by less than $3.7\cdot10^{-5}$, which is why a search over graphs of small order does not distinguish them. The equality statement fails as well, since the extremal graphs are not only the complete bipartite ones.

math.CO↗

arXiv2026

The typical structure of oriented graphs and digraphs with forbidden blow-up of transitive tournaments

For integers $r\ge 2$, $t\ge 1$ and a real number $a\in(3/2,2]$, we study the typical structure of oriented graphs and digraphs that do not contain a blow-up $T_{r+1}^t$ of a transitive tournament. We prove that almost every $T_{r+1}^t$-free oriented graph on n vertices admits an r-partition $V_1\cup\cdots\cup V_r$ such that each induced subgraph $G[V_i]$ is $T_2^t$-free, and the same holds for almost every $T_{r+1}^t$-free digraph.Consequently, the number $f(n,T_{r+1}^t)$ of labelled $T_{r+1}^t$-free oriented graphs satisfies $f(n,T_{r+1}^t)=|\mathcal{P}_{n,r,t}|(1+o(1))$, where $\mathcal{P}_{n,r,t}$ is the family of oriented graphs admitting such an r-partition with each part $T_2^t$-free; an analogous statement holds for digraphs.When $t=1$ this recovers the result of K"uhn, Osthus, Townsend and Zhao (2017) that almost all $T_{r+1}$-free oriented graphs (resp. digraphs) are r-partite, thereby confirming a generalised form of Cherlin's conjecture. Our proof combines the hypergraph container method, a weighted Erd\H{o}s-Stone theorem, and a stability analysis for near-extremal $T_{r+1}^t$-free digraphs.

math.CO↗

arXiv2026

A container theorem for general digraphs with forbidden subdigraphs

In a seminal work, K\"uhn, Osthus, Townsend, and Zhao used the hypergraph container method to determine the typical structure of oriented graphs and digraphs avoiding a fixed tournament or cycle. Their main tool, a container theorem for oriented graphs, does not directly extend to all digraphs due to the existence of counterexamples such as the double triangle $DK_3$. In this paper we prove a container theorem for general digraphs under a natural sparsity condition. For the edge-weight parameter $a=2$, this condition permits digraphs with $2$-cycles (density at most $1$) but excludes denser obstructions like $DK_3$; for larger $a$ it allows digraphs with a controlled density of $2$-cycles. As applications, we obtain asymptotic counting results for $H$-free digraphs and describe the typical structure of digraphs avoiding a fixed digraph $H$ satisfying our condition. Our results unify and extend several previous results in the area.

math.CO↗

arXiv2025

World In Your Hands: A Large-Scale and Open-Source Ecosystem for Learning Human-Centric Manipulation in the Wild

We introduce World In Your Hands (WIYH), a large-scale open-source ecosystem comprising over 1,000 hours of human manipulation data collected in-the-wild with millimeter-scale motion accuracy. Specifically, WIYH includes (1) the Oracle Suite, a wearable data collection kit with an auto-labeling pipeline for accurate motion capture; (2) the WIYH Dataset, featuring over 1,000 hours of multimodal manipulation data across hundreds of skills in diverse real-world scenarios; and (3) extensive annotations and benchmarks supporting tasks from perception to action. Furthermore, experiments based on the WIYH ecosystem show that integrating WIYH's human-centric data improves robotic manipulation success rates from 8% to 60% in cluttered scenes. World In Your Hands provides a foundation for advancing human-centric data collection and cross-embodiment policy learning. All data and hardware design will be open-source.

cs.RO↗

arXiv2025

Spectral extrema of graphs of given even size forbidding H(4,3)

A graph is sad to be $H$-free if it does not contain $H$ as a subgraph. Let $H(k,3)$ be the graph formed by taking a cycle of length $k$ and a triangle on a common vertex. Li, Lu and Peng [Discrete Math. 346 (2023) 113680] proved that if $G$ is an $H(3,3)$-free graph of size $m \geq 8$, then the spectral radius $\rho(G) \leq \frac{1+\sqrt{4 m-3}}{2}$ with equality if and only if $G \cong S_{\frac{m+3}{2}, 2}$, where $S_{\frac{m+3}{2}, 2}=K_2 \vee \frac{m-1}{2}K_1$. Note that the bound is attainable only when $m$ is odd. Recently, Pirzada and Rehman [Comput. Appl. Math. 44 (2025) 295] proved that if $G$ is an $\{H(3,3),H(4,3)\}$-free graph of even size $m \geq 10$, then $\rho(G) \leq \rho^{\prime}(m)$ with equality if and only if $G \cong S_{\frac{m+4}{2}, 2}^{-}$, where $\rho^{\prime}(m)$ is the largest root of $x^4-m x^2-(m-2) x+\frac{m}{2}-1=0$, and $S_{\frac{m+4}{2}, 2}^{-}$ is the graph obtained from $S_{\frac{m+4}{2}, 2}$ by deleting an edge incident to a vertex of degree two. In this paper, we improve the result of Pirzada and Rehman by showing that if $G$ is an $H(4,3)$-free graph of even size $m \geq 38$ without isolated vertices, then $\rho(G) \leq \rho^{\prime}(m)$ with equality if and only if $G \cong S_{\frac{m+4}{2}, 2}^{-}$.

math.CO↗

arXiv2023

Extremal trees, unicyclic and bicyclic graphs with respect to $p$-Sombor spectral radii

For a graph $G=(V,E)$ and $v_{i}\in V$, denote by $d_{v_{i}}$ (or $d_{i}$ for short) the degree of vertex $v_{i}$. The $p$-Sombor matrix $\textbf{S}_{\textbf{p}}(G)$ ($p\neq0$) of a graph $G$ is a square matrix, where the $(i,j)$-entry is equal to $\displaystyle (d_{i}^{p}+d_{j}^{p})^{\frac{1}{p}}$ if the vertices $v_{i}$ and $v_{j}$ are adjacent, and 0 otherwise. The $p$-Sombor spectral radius of $G$, denoted by $\displaystyle \rho(\textbf{S}_{\textbf{p}}(G))$, is the largest eigenvalue of the $p$-Sombor matrix $\textbf{S}_{\textbf{p}}(G)$. In this paper, we consider the extremal trees, unicyclic and bicyclic graphs with respect to the $p$-Sombor spectral radii. We characterize completely the extremal graphs with the first three maximum Sombor spectral radii, which answers partially a problem posed by Liu et al. in [MATCH Commun. Math. Comput. Chem. 87 (2022) 59-87].

math.CO↗

arXiv2023

Two spectral extremal results for graphs with given order and rank

The spectral radius and rank of a graph are defined to be the spectral radius and rank of its adjacency matrix, respectively. It is an important problem in spectral extremal graph theory to determine the extremal graph that has the maximum or minimum spectral radius over certain families of graphs. Monsalve and Rada [Extremal spectral radius of graphs with rank 4, Linear Algebra Appl. 609 (2021) 1-11] obtained the extremal graphs with maximum and minimum spectral radii among all graphs with order n and rank 4. In this paper, we first determine the extremal graph which attains the maximum spectral radius among all graphs with any given order n and rank r, and further determine the extremal graph which attains the minimum spectral radius among all graphs with order n and rank 5.

math.CO↗

arXiv2022

Extremal trees with respect to spectral radius of restrictedly weighted adjacency matrices

For a graph $G=(V,E)$ and $v_{i}\in V$, denote by $d_{i}$ the degree of vertex $v_{i}$. Let $f(x, y)>0$ be a real symmetric function in $x$ and $y$. The weighted adjacency matrix $A_{f}(G)$ of a graph $G$ is a square matrix, where the $(i,j)$-entry is equal to $\displaystyle f(d_{i}, d_{j})$ if the vertices $v_{i}$ and $v_{j}$ are adjacent and 0 otherwise. Li and Wang \cite{U9} tried to unify methods to study spectral radius of weighted adjacency matrices of graphs weighted by various topological indices. If $\displaystyle f'_{x}(x, y)\geq0$ and $\displaystyle f''_{x}(x, y)\geq0$, then $\displaystyle f(x, y)$ is said to be increasing and convex in variable $x$, respectively. They obtained the tree with the largest spectral radius of $A_{f}(G)$ is a star or a double star when $f(x, y)$ is increasing and convex in variable $x$. In this paper, we add the following restriction: $f(x_{1},y_{1})\geq f(x_{2},y_{2})$ if $x_{1}+y_{1}=x_{2}+y_{2}$ and $\mid x_{1}-y_{1}\mid>\mid x_{2}-y_{2}\mid$ and call $A_f(G)$ the restrictedly weighted adjacency matrix of $G$. The restrictedly weighted adjacency matrix contains weighted adjacency matrices weighted by first Zagreb index, first hyper-Zagreb index, general sum-connectivity index, forgotten index, Somber index, $p$-Sombor index and so on. We obtain the extremal trees with the smallest and the largest spectral radius of $A_{f}(G)$. Our results push ahead Li and Wang's research on unified approaches.

math.CO↗

arXiv2022

Arithmetic-Geometric spectral radii of Unicyclic graphs

Let $d_{v_{i}}$ be the degree of the vertex $v_{i}$ of $G$. The arithmetic-geometric matrix $A_{ag}(G)$ of a graph $G$ is a square matrix, where the $(i,j)$-entry is equal to $\displaystyle \frac{d_{v_{i}}+d_{v_{j}}}{2\sqrt{d_{v_{i}}d_{v_{j}}}}$ if the vertices $v_{i}$ and $v_{j}$ are adjacent, and 0 otherwise. The arithmetic-geometric spectral radius of $G$, denoted by $\rho_{ag}(G)$, is the largest eigenvalue of the arithmetic-geometric matrix $A_{ag}(G)$. In this paper, the unicyclic graphs of order $n\geq5$ with the smallest and first four largest arithmetic-geometric spectral radii are determined.

math.SP↗

arXiv2021

Arithmetic-Geometric Spectral Radius of Trees and Unicyclic Graphs

The arithmetic-geometric matrix $A_{ag}(G)$ of a graph $G$ is a square matrix, where the $(i,j)$-entry is equal to $\displaystyle \frac{d_{i}+d_{j}}{2\sqrt{d_{i}d_{j}}}$ if the vertices $v_{i}$ and $v_{j}$ are adjacent, and 0 otherwise. The arithmetic-geometric spectral radius of $G$, denoted by $\rho_{ag}(G)$, is the largest eigenvalue of the arithmetic-geometric matrix $A_{ag}(G)$. Let $S_{n}$ be the star of order $n\geq3$ and $S_{n}+e$ be the unicyclic graph obtained from $S_{n}$ by adding an edge. In this paper, we prove that for any tree $T$ of order $n\geq2$, $\displaystyle 2\cos\frac{\pi}{n+1}\leq\rho_{ag}(P_{n})\leq\rho_{ag}(T)\leq\rho_{ag}(S_{n})=\frac{n}{2},$ with equality if and only if $T\cong P_{n}$ for the lower bound, and if and only if $T\cong S_{n}$ for the upper bound. We also prove that for any unicyclic graph $G$ of order $n\geq3$, $\displaystyle 2=\rho_{ag}(C_{n})\leq\rho_{ag}(G)\leq\rho_{ag}(S_{n}+e),$ the lower (upper, respectively) bound is attained if and only if $T\cong C_{n}$ ($T\cong S_{n}+e$, respectively) and $\displaystyle\rho_{ag}(S_{n}+e)<\frac{n}{2}$ for $n\geq7$.

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