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Ligong Wang

Publications and source records attributed to Ligong Wang.

At least 91 records · Page 5Linked to original sources
arXiv2018

Edge Connectivity, Packing Spanning Trees, and Eigenvalues of Graphs

Let $\mathcal{G}$ be the set of simple graphs (or multigraphs) $G$ such that for each $G \in \mathcal{G}$ there exists at least two non-empty disjoint proper subsets $V_{1},V_{2}\subseteq V(G)$ satisfying $V(G)\setminus(V_{1} \cup V_{2})\neq ϕ$ and edge connectivity $κ'(G)=e(V_{i},V(G)\backslash V_{i})$ for $1\leq i \leq 2$. A multigraph is a graph with possible multiple edges, but no loops. Let $τ(G)$ be the maximum number of edge-disjoint spanning trees of a graph $G$. Motivated by a question of Seymour on the relationship between eigenvalues of a graph $G$ and bounds of $τ(G)$, we mainly give the relationship between the third largest (signless Laplacian) eigenvalue and the bound of $κ'(G)$ and $τ(G)$ of a simple graph or a multigraph $G\in\mathcal{G}$, respectively.

math.CO↗
arXiv2018

Gallai-Ramsey numbers for a class of graphs with five vertices

Given two graphs $G$ and $H$, the $k$-colored Gallai-Ramsey number $gr_k(G : H)$ is defined to be the minimum integer $n$ such that every $k$-coloring of the complete graph on $n$ vertices contains either a rainbow copy of $G$ or a monochromatic copy of $H$. In this paper, we consider $gr_k(K_3 : H)$ where $H$ is a connected graph with five vertices and at most six edges. There are in total thirteen graphs in this graph class, and the Gallai-Ramsey numbers for some of them have been studied step by step in several papers. We determine all the Gallai-Ramsey numbers for the remaining graphs, and we also obtain some related results for a class of unicyclic graphs.

math.CO↗
arXiv2018

Merging the A- and Q-spectral theories for digraphs

Let $G$ be a digraph and $A(G)$ be the adjacency matrix of $G$. Let $D(G)$ be the diagonal matrix with outdegrees of vertices of $G$. For any real $α\in[0,1]$, Liu et al. \cite{LWCL} defined the matrix $A_α(G)$ as $$A_α(G)=αD(G)+(1-α)A(G).$$ The largest modulus of the eigenvalues of $A_α(G)$ is called the $A_α$ spectral radius of $G$. In this paper, we determine the digraphs which attain the maximum (or minimum) $A_α$ spectral radius among all strongly connected digraphs with given parameters such as girth, clique number, vertex connectivity or arc connectivity. We also discuss a number of open problems.

math.CO↗
arXiv2017

The rank of a complex unit gain graph in terms of the rank of its underlying graph

Let $Φ=(G, φ)$ be a complex unit gain graph (or $\mathbb{T}$-gain graph) and $A(Φ)$ be its adjacency matrix, where $G$ is called the underlying graph of $Φ$. The rank of $Φ$, denoted by $r(Φ)$, is the rank of $A(Φ)$. Denote by $θ(G)=|E(G)|-|V(G)|+ω(G)$ the dimension of cycle spaces of $G$, where $|E(G)|$, $|V(G)|$ and $ω(G)$ are the number of edges, the number of vertices and the number of connected components of $G$, respectively. In this paper, we investigate bounds for $r(Φ)$ in terms of $r(G)$, that is, $r(G)-2θ(G)\leq r(Φ)\leq r(G)+2θ(G)$, where $r(G)$ is the rank of $G$. As an application, we also prove that $1-θ(G)\leq\frac{r(Φ)}{r(G)}\leq1+θ(G)$. All corresponding extremal graphs are characterized.

math.CO↗
arXiv2017

Capacity Results on Multiple-Input Single-Output Wireless Optical Channels

This paper derives upper and lower bounds on the capacity of the multiple-input single-output free-space optical intensity channel with signal-independent additive Gaussian noise subject to both an average-intensity and a peak-intensity constraint. In the limit where the signal-to-noise ratio (SNR) tends to infinity, the asymptotic capacity is specified, while in the limit where the SNR tends to zero, the exact slope of the capacity is also given.

cs.IT↗
arXiv2017

Covert Communication with Channel-State Information at the Transmitter

We consider the problem of covert communication over a state-dependent channel, where the transmitter has causal or noncausal knowledge of the channel states. Here, "covert" means that a warden on the channel should observe similar statistics when the transmitter is sending a message and when it is not. When a sufficiently long secret key is shared between the transmitter and the receiver, we derive closed-form formulas for the maximum achievable covert communication rate ("covert capacity") for discrete memoryless channels and, when the transmitter's channel-state information (CSI) is noncausal, for additive white Gaussian noise (AWGN) channels. For certain channel models, including the AWGN channel, we show that the covert capacity is positive with CSI at the transmitter, but is zero without CSI. We also derive lower bounds on the rate of the secret key that is needed for the transmitter and the receiver to achieve the covert capacity.

cs.IT↗
arXiv2017

The minimum Q-index of strongly connected bipartite digraphs with complete bipartite subdigraphs

Let $\mathcal{G}_{n,p,q}$ denote the set of strongly connected bipartite digraphs on $n$ vertices which contain a complete bipartite subdigraph $\overleftrightarrow{K_{p,q}}$, where $p, q, n$ are positive integers and $p+q \leq n$. In this paper, we study the Q-index (i.e. the signless Laplacian spectral radius) of $\mathcal{G}_{n,p,q}$, and determine the extremal digraph that has the minimum Q-index.

math.CO↗
arXiv2017

Ordering of bicyclic graphs by matching energy

Let $G$ be a simple graph of order $n$ and $μ_{1},μ_{2},\ldots,μ_{n}$ be the roots of its matching polynomial. The matching energy is defined as the sum $\sum^{n}_{i=1}|μ_{i}|$, which was introduced by Gutman and Wagner in 2012. In this paper, the graphs with the first five smallest matching energies among all bicyclic graphs for order $n>5$ are determined.

math.CO↗
arXiv2017

Optimal Throughput for Covert Communication Over a Classical-Quantum Channel

This paper considers the problem of communication over a memoryless classical-quantum wiretap channel subject to the constraint that the eavesdropper on the channel should not be able to learn whether the legitimate parties are using the channel to communicate or not. Specifically, the relative entropy between the output quantum states at the eavesdropper when a codeword is transmitted and when no input is provided must be sufficiently small. Extending earlier works, this paper proves the "square-root law" for a broad class of classical-quantum channels: the maximum amount of information that can be reliably and covertly transmitted over $n$ uses of such a channel scales like $\sqrt{n}$. The scaling constant is also determined.

cs.IT↗
arXiv2017

A modified generalized shift-splitting preconditioner for nonsymmetric saddle point problems

For the nonsymmetric saddle point problems with nonsymmetric positive definite (1,1) parts, the modified generalized shift-splitting (MGSSP) preconditioner as well as the MGSSP iteration method are derived in this paper, which generalize the MSSP preconditioner and the MSSP iteration method newly developed by Huang and Su (J. Comput. Appl. Math. 2017), respectively. The convergent and semi-convergent analysis of the MGSSP iteration method are presented, and we prove that this method is unconditionally convergent and semi-convergent. In addition, some spectral properties of the preconditioned matrix are carefully analyzed. Numerical results demonstrate the robustness and effectiveness of the MGSSP preconditioner and the MGSSP iteration method, and also illustrate that the MGSSP iteration method outperforms the GSS and GMSS iteration methods, and the MGSSP preconditioner is superior to the shift-splitting (SS), generalized SS (GSS), modified SS (MSS) and generalized MSS (GMSS) preconditioners for the GMRES method for solving the nonsymmetric saddle point problems.

math.NA↗
arXiv2016

Skew-rank of an oriented graph in terms of the rank and dimension of cycle space of its underlying graph

Let $G^σ$ be an oriented graph and $S(G^σ)$ be its skew-adjacency matrix, where $G$ is called the underlying graph of $G^σ$. The skew-rank of $G^σ$, denoted by $sr(G^σ)$, is the rank of $S(G^σ)$. Denote by $d(G)=|E(G)|-|V(G)|+θ(G)$ the dimension of cycle spaces of $G$, where $|E(G)|$, $|V(G)|$ and $θ(G)$ are the edge number, vertex number and the number of connected components of $G$, respectively. Recently, Wong, Ma and Tian [European J. Combin. 54 (2016) 76--86] proved that $sr(G^σ)\leq r(G)+2d(G)$ for an oriented graph $G^σ$, where $r(G)$ is the rank of the adjacency matrix of $G$, and characterized the graphs whose skew-rank attain the upper bound. However, the problem of the lower bound of $sr(G^σ)$ of an oriented graph $G^σ$ in terms of $r(G)$ and $d(G)$ of its underlying graph $G$ is left open till now. In this paper, we prove that $sr(G^σ)\geq r(G)-2d(G)$ for an oriented graph $G^σ$ and characterize the graphs whose skew-rank attain the lower bound.

math.CO↗
arXiv2016

Upper bounds on the Q-spectral radius of book-free and/or $K_{s,t}$-free graphs

In this paper, we prove two results about the signless Laplacian spectral radius $q(G)$ of a graph $G$ of order $n$ with maximum degree $Δ$. Let $B_{n}=K_{2}+\overline{K_{n}}$ denote a book, i.e., the graph $B_{n}$ consists of $n$ triangles sharing an edge. (1) Let $1< k\leq l< Δ< n$ and $G$ be a connected \{$B_{k+1},K_{2,l+1}$\}-free graph of order $n$ with maximum degree $Δ$. Then $$\displaystyle q(G)\leq \frac{1}{4}[3Δ+k-2l+1+\sqrt{(3Δ+k-2l+1)^{2}+16l(Δ+n-1)}.$$ with equality holds if and only if $G$ is a strongly regular graph with parameters ($Δ$, $k$, $l$). (2) Let $s\geq t\geq 3$, and let $G$ be a connected $K_{s,t}$-free graph of order $n$ $(n\geq s+t)$. Then $$q(G)\leq n+(s-t+1)^{1/t}n^{1-1/t}+(t-1)(n-1)^{1-3/t}+t-3.$$

math.CO↗
arXiv2016

On the signless Laplacian spectral radius of $C_{4}$-free $k$-cyclic graphs

A $k$-cyclic graph is a connected graph of order $n$ and size $n+k-1$. In this paper, we determine the maximal signless Laplacian spectral radius and the corresponding extremal graph among all $C_{4}$-free $k$-cyclic graphs of order $n$. Furthermore, we determine the first three unicyclic, and bicyclic, $C_{4}$-free graphs whose spectral radius of the signless Laplacian is maximal. Similar results are obtained for the (combinatorial) Laplacian.

math.CO↗
arXiv2016

Hermitian-Randić matrix and Hermitian-Randić energy of mixed graphs

Let $M$ be a mixed graph and $H(M)$ be its Hermitian-adjacency matrix. If we add every edge and arc in $M$ a Randić weight, then we can get a new weighted Hermitian-adjacency matrix. What are the properties of this new matrix? Motivated by this, we define the Hermitian-Randić matrix $R_{H}(M)=(r_{h})_{kl}$ of a mixed graph $M$, where $(r_{h})_{kl}=-(r_{h})_{lk}=\frac{\textbf{i}}{\sqrt{d_{k}d_{l}}}$ ($\textbf{i}=\sqrt{-1}$) if $(v_{k},v_{l})$ is an arc of $M$, $(r_{h})_{kl}=(r_{h})_{lk}=\frac{1}{\sqrt{d_{k}d_{l}}}$ if $v_{k}v_{l}$ is an undirected edge of $M$, and $(r_{h})_{kl}=0$ otherwise. In this paper, firstly, we compute the characteristic polynomial of the Hermitian-Randić matrix of a mixed graph. Furthermore, we give bounds to the Hermitian-Randić energy of a general mixed graph. Finally, we give some results about the Hermitian-Randić energy of mixed trees.

math.CO↗
arXiv2016

The signless Laplacian spectral radius of subgraphs of regular graphs

Let $q(H)$ be the signless Laplacian spectral radius of a graph $H$. In this paper, we prove that \\1. Let $H$ be a proper subgraph of a $Δ$-regular graph $G$ with $n$ vertices and diameter $D$. Then $$2Δ- q(H)>\frac{1}{n(D-\frac{1}{4})}.$$ \\2. Let $H$ be a proper subgraph of a $k$-connected $Δ$-regular graph $G$ with $n$ vertices, where $k\geq 2$. Then $$2Δ-q(H)>\frac{2(k-1)^{2}}{2(n-Δ)(n-Δ+2k-4)+(n+1)(k-1)^{2}}.$$ Finally, we compare the two bounds. We obtain that when $k>2\sqrt{\frac{(n-Δ)(n+Δ-4)}{n(4D-3)-2}}+1$, the second bound is always better than the first. On the other hand, when $k<\frac{2(n-Δ)}{\sqrt{n(4D-3)-2}}+1$, the first bound is always better than the second.

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