arXiv subjects
Publications and source records attributed to Ruifang Liu.
Let $G$ be a non-bipartite graph which does not contain any odd cycle of length at most $2k+1$. In this paper, we determine the maximum $Q$-index of $G$ if its order is fixed, and the corresponding extremal graph is uniquely characterized. Moreover, if the size of $G$ is given, the maximum $Q$-index of $G$ and the unique extremal graph are also proved.
For integer $k\geq2,$ a spanning $k$-ended-tree is a spanning tree with at most $k$ leaves. Motivated by the closure theorem of Broersma and Tuinstra [Independence trees and Hamilton cycles, J. Graph Theory 29 (1998) 227--237], we provide tight spectral conditions to guarantee the existence of a spanning $k$-ended-tree in a connected graph of order $n$ with extremal graphs being characterized. Moreover, by adopting Kaneko's theorem [Spanning trees with constraints on the leaf degree, Discrete Appl. Math. 115 (2001) 73--76], we also present tight spectral conditions for the existence of a spanning tree with leaf degree at most $k$ in a connected graph of order $n$ with extremal graphs being determined, where $k\geq1$ is an integer.
For a graph $G$ with domination number $γ$, Hedetniemi, Jacobs and Trevisan [European Journal of Combinatorics 53 (2016) 66-71] proved that $m_{G}[0,1)\leq γ$, where $m_{G}[0,1)$ means the number of Laplacian eigenvalues of $G$ in the interval $[0,1)$. Let $T$ be a tree with diameter $d$. In this paper, we show that $m_{T}[0,1)\geq (d+1)/3$. However, such a lower bound is false for general graphs. All trees achieving the lower bound are completely characterized. Moreover, for a tree $T$, we establish a relation between the Laplacian eigenvalues, the diameter and the domination number by showing that the domination number of $T$ is equal to $(d+1)/3$ if and only if it has exactly $(d+1)/3$ Laplacian eigenvalues less than one. As an application, it also provides a new type of trees, which show the sharpness of an inequality due to Hedetniemi, Jacobs and Trevisan.
The difference between the two largest eigenvalues of the adjacency matrix of a graph $G$ is called the spectral gap of $G.$ If $G$ is a regular graph, then its spectral gap is equal to algebraic connectivity. Abdi, Ghorbani and Imrich, in [European J. Combin. 95 (2021) 103328], showed that the minimum algebraic connectivity of cubic connected graphs on $2n$ vertices is $(1+o(1))\frac{π^{2}}{2n^{2}}$, which is attained on non-bipartite graphs. Motivated by the above result, we in this paper investigate the algebraic connectivity of cubic bipartite graphs. We prove that the minimum algebraic connectivity of cubic bipartite graphs on $2n$ vertices is $(1+o(1))\frac{π^{2}}{n^{2}}$. Moreover, the unique cubic bipartite graph with minimum algebraic connectivity is completed characterized. Based on the relation between the algebraic connectivity and spectral gap of regular graphs, the cubic bipartite graph with minimum spectral gap and the corresponding asymptotic value are also presented. In [J. Graph Theory 99 (2022) 671--690], Horak and Kim established a sharp upper bound for the number of perfect matchings in terms of the Fibonacci number. We obtain a spectral characterization for the extremal graphs by showing that a cubic bipartite graph has the maximum number of perfect matchings if and only if it minimizes the algebraic connectivity.
For integer $k\geq2,$ a graph $G$ is called $k$-leaf-connected if $|V(G)|\geq k+1$ and given any subset $S\subseteq V(G)$ with $|S|=k,$ $G$ always has a spanning tree $T$ such that $S$ is precisely the set of leaves of $T.$ Thus a graph is $2$-leaf-connected if and only if it is Hamilton-connected. In this paper, we present a best possible condition based upon the size to guarantee a graph to be $k$-leaf-connected, which not only improves the results of Gurgel and Wakabayashi [On $k$-leaf-connected graphs, J. Combin. Theory Ser. B 41 (1986) 1-16] and Ao, Liu, Yuan and Li [Improved sufficient conditions for $k$-leaf-connected graphs, Discrete Appl. Math. 314 (2022) 17-30], but also extends the result of Xu, Zhai and Wang [An improvement of spectral conditions for Hamilton-connected graphs, Linear Multilinear Algebra, 2021]. Our key approach is showing that an $(n+k-1)$-closed non-$k$-leaf-connected graph must contain a large clique if its size is large enough. As applications, sufficient conditions for a graph to be $k$-leaf-connected in terms of the (signless Laplacian) spectral radius of $G$ or its complement are also presented.
Let $λ^{*}$ be the maximum spectral radius of connected irregular graphs on $n$ vertices with maximum degree $Δ$. Liu, Shen and Wang (2007) conjectured that $\lim_{n\rightarrow \infty}(n^{2}(Δ-λ^{*}))/(Δ-1)=π^{2},$ which describes the asymptotic behavior for the maximum spectral radius of irregular graphs. Focusing on this conjecture, we consider the maximum spectral radius of connected subcubic bipartite graphs. The unique connected subcubic bipartite graph with the maximum spectral radius is determined. Let $G$ be a $k$-connected irregular graph with spectral radius $λ_{1}(G)$, we present a lower bound for $Δ-λ_{1}(G)$. Moreover, if $H$ is a proper subgraph of a $k$-connected $Δ$-regular graph, a lower bound for $Δ-λ_{1}(H)$ is also obtained. These bounds improve some previous results.
A classic result in extremal graph theory, known as Mantel's theorem, states that every non-bipartite graph of order $n$ with size $m>\lfloor \frac{n^{2}}{4}\rfloor$ contains a triangle. Lin, Ning and Wu [Comb. Probab. Comput. 30 (2021) 258-270] proved a spectral version of Mantel's theorem for given order $n.$ Zhai and Shu [Discrete Math. 345 (2022) 112630] investigated a spectral version for fixed size $m.$ In this paper, we prove $Q$-spectral versions of Mantel's theorem.
A bipartite graph is subcubic if it is an irregular bipartite graph with maximum degree three. In this paper, we prove that the asymptotic value of maximum spectral radius over subcubic bipartite graphs of order $n$ is $3-\varTheta(\frac{π^{2}}{n^{2}})$. Our key approach is taking full advantage of the eigenvalues of certain tridiagonal matrices, due to Willms [SIAM J. Matrix Anal. Appl. 30 (2008) 639--656]. Moreover, for large maximum degree, i.e., the maximum degree is at least $\lfloor n/2 \rfloor$, we characterize irregular bipartite graphs with maximum spectral radius. For general maximum degree, we present an upper bound on the spectral radius of irregular bipartite graphs in terms of the order and maximum degree.
In multi-label text classification (MLTC), each given document is associated with a set of correlated labels. To capture label correlations, previous classifier-chain and sequence-to-sequence models transform MLTC to a sequence prediction task. However, they tend to suffer from label order dependency, label combination over-fitting and error propagation problems. To address these problems, we introduce a novel approach with multi-task learning to enhance label correlation feedback. We first utilize a joint embedding (JE) mechanism to obtain the text and label representation simultaneously. In MLTC task, a document-label cross attention (CA) mechanism is adopted to generate a more discriminative document representation. Furthermore, we propose two auxiliary label co-occurrence prediction tasks to enhance label correlation learning: 1) Pairwise Label Co-occurrence Prediction (PLCP), and 2) Conditional Label Co-occurrence Prediction (CLCP). Experimental results on AAPD and RCV1-V2 datasets show that our method outperforms competitive baselines by a large margin. We analyze low-frequency label performance, label dependency, label combination diversity and coverage speed to show the effectiveness of our proposed method on label correlation learning.
Brualdi and Hoffman (1985) proposed the problem of determining the maximal spectral radius of graphs with given size. In this paper, we consider the Brualdi-Hoffman type problem of graphs with given matching number. The maximal $Q$-spectral radius of graphs with given size and matching number is obtained, and the corresponding extremal graphs are also determined.
A fractional matching of a graph $G$ is a function $f:E(G) \to [0,1]$ such that for any $v\in V(G)$, $\sum_{e\in E_G(v)}f(e)\leq 1$ where $E_G(v) = \{e \in E(G): e$ is incident with $v$ in $G\}$. The fractional matching number of $G$ is $μ_{f}(G) = \max\{\sum_{e\in E(G)} f(e): f$ is fractional matching of $G\}$. For any real numbers $a \ge 0$ and $k \in (0, n)$, it is observed that if $n = |V(G)|$ and $δ(G) > \frac{n-k}{2}$, then $μ_{f}(G)>\frac{n-k}{2}$. We determine a function $φ(a, n,δ, k)$ and show that for a connected graph $G$ with $n = |V(G)|$, $δ(G) \leq\frac{n-k}{2}$, spectral radius $λ_1(G)$ and complement $\overline{G}$, each of the following holds. (i) If $λ_{1}(aD(G)+A(G))<φ(a, n, δ, k),$ then $μ_{f}(G)>\frac{n-k}{2}.$ (ii) If $λ_{1}(aD(\overline{G})+A(\overline{G}))<(a+1)(δ+k-1),$ then $μ_{f}(G)>\frac{n-k}{2}.$ As corollaries, sufficient spectral condition for fractional perfect matchings and analogous results involving $Q$-index and $A_α$-spectral radius are obtained, and former spectral results in [European J. Combin. 55 (2016) 144-148] are extended.
Let $q(G)$ denote the $Q$-index of a graph $G$, which is the largest signless Laplacian eigenvalue of $G$. We prove best possible upper bounds of $q(G)$ and best possible lower bounds of $q(\overline{G})$ for a connected graph $G$ to be $k$-connected and maximally connected, respectively. Similar upper bounds of $q(G)$ and lower bounds of $q(\overline{G})$ to assure $G$ to be super-connected are also obtained. Upper bounds of $q(G)$ and lower bounds of $q(\overline{G})$ to assure a connected triangle-free graph $G$ to be $k$-connected, maximally connected and super-connected are also respectively investigated.
We propose Neural Entity Reasoner (NE-Reasoner), a framework to introduce global consistency of recognized entities into Neural Reasoner over Named Entity Recognition (NER) task. Given an input sentence, the NE-Reasoner layer can infer over multiple entities to increase the global consistency of output labels, which then be transfered into entities for the input of next layer. NE-Reasoner inherits and develops some features from Neural Reasoner 1) a symbolic memory, allowing it to exchange entities between layers. 2) the specific interaction-pooling mechanism, allowing it to connect each local word to multiple global entities, and 3) the deep architecture, allowing it to bootstrap the recognized entity set from coarse to fine. Like human beings, NE-Reasoner is able to accommodate ambiguous words and Name Entities that rarely or never met before. Despite the symbolic information the model introduced, NE-Reasoner can still be trained effectively in an end-to-end manner via parameter sharing strategy. NE-Reasoner can outperform conventional NER models in most cases on both English and Chinese NER datasets. For example, it achieves state-of-art on CoNLL-2003 English NER dataset.
Let $λ_{1}(G)$ and $μ_{1}(G)$ denote the spectral radius and the Laplacian spectral radius of a graph $G$, respectively. Li in [Electronic J. Linear Algebra 34 (2018) 389-392] proved sharp upper bounds of $λ_{1}(G)$ based on the connectivity to assure a connected graph to be Hamiltonian and traceable, respectively. In this paper, we present best possible upper bounds of $λ_{1}(G)$ for $k$-connected graphs to be Hamiltonian-connected and homogeneously traceable, respectively. Furthermore, best possible upper bounds of $μ_{1}(G)$ to predict $k$-connected graphs to be Hamiltonian-connected, Hamiltonian and traceable are originally proved, respectively.
Let $τ(G)$ and $κ'(G)$ denote the edge-connectivity and the spanning tree packing number of a graph $G$, respectively. Proving a conjecture initiated by Cioaba and Wong, Liu et al. in 2014 showed that for any simple graph $G$ with minimum degree $δ\ge 2k \ge 4$, if the second largest adjacency eigenvalue of $G$ satisfies $λ_2(G) < δ- \frac{2k-1}{δ+1}$, then $τ(G) \ge k$. Similar results involving the Laplacian eigenvalues and the signless Laplacian eigenvalues of $G$ are also obtained. In this paper, we find a function $f(δ, k, g)$ such that for every graph $G$ with minimum degree $δ\ge 2k \ge 4$ and girth $g \ge 3$, if its second largest adjacency eigenvalue satisfies $λ_2(G) < f(δ, k, g)$, then $τ(G) \ge k$. As $f(δ, k, 3) = δ- \frac{2k-1}{δ+1}$, this extends the above-mentioned result of Liu et al. Related results involving the girth of the graph, Laplacian eigenvalues and the signless Laplacian eigenvalues to describe $τ(G)$ and $κ'(G)$ are also obtained.
A {\it fractional matching} of a graph $G$ is a function $f$ giving each edge a number in $[0,1]$ so that $\sum_{e\in Γ(v)}f(e)\leq 1$ for each $v\in V(G)$, where $Γ(v)$ is the set of edges incident to $v$. The {\it fractional matching number} of $G$, written $α'_{*}(G)$, is the maximum of $\sum_{e\in E(G)}f(e)$ over all fractional matchings $f$. In this paper, we propose the relations between the fractional matching number and the signless Laplacian spectral radius of a graph. As applications, we also give sufficient spectral conditions for existence of a fractional perfect matching in a graph in terms of the signless Laplacian spectral radius of the graph and its complement.
Let $G$ be a connected graph with vertex set $V(G)=\{v_{1},v_{2},...,v_{n}\}$. The distance matrix $D(G)=(d_{ij})_{n\times n}$ is the matrix indexed by the vertices of $G,$ where $d_{ij}$ denotes the distance between the vertices $v_{i}$ and $v_{j}$. Suppose that $λ_{1}(D)\geqλ_{2}(D)\geq\cdots\geqλ_{n}(D)$ are the distance spectrum of $G$. The graph $G$ is said to be determined by its $D$-spectrum if with respect to the distance matrix $D(G)$, any graph having the same spectrum as $G$ is isomorphic to $G$. In this paper, we give the distance characteristic polynomial of some graphs with small diameter, and also prove that these graphs are determined by their $D$-spectra.
Let $G$ be a simple connected graph of order $n$ and $D(G)$ be the distance matrix of $G.$ Suppose that $λ_{1}(D(G))\geqλ_{2}(D(G))\geq\cdots\geqλ_{n}(D(G))$ are the distance spectrum of $G$. A graph $G$ is said to be determined by its $D$-spectrum if with respect to the distance matrix $D(G)$, any graph with the same spectrum as $G$ is isomorphic to $G$. In this paper, we consider spectral characterization on the second largest distance eigenvalue $λ_{2}(D(G))$ of graphs, and prove that the graphs with $λ_{2}(D(G))\leq\frac{17-\sqrt{329}}{2}\approx-0.5692$ are determined by their $D$-spectra.