arXiv subjects

Ligong Wang

Publications and source records attributed to Ligong Wang.

At least 73 records · Page 4Linked to original sources

arXiv2022

The $A_α$ spectral radius with given independence number $n-4$

Let $G$ be a graph with adjacency matrix $A(G)$ and degree diagonal matrix $D (G)$. In 2017, Nikiforov [Appl. Anal. Discrete Math., 11 (2017) 81--107] defined the matrix $A_α(G) = αD(G) + (1-α)A(G)$ for any real $α\in[0,1]$. The largest eigenvalue of $A(G)$ is called the spectral radius of $G$, while the largest eigenvalue of $A_α(G)$ is called the $A_α$ spectral radius of $G$. Let $\mathcal{G}_{n,i}$ be the set of graphs of order $n$ with independence number $i$. Recently, for all graphs in $\mathcal{G}_{n,i}$ having the minimum or the maximum $A$, $Q$ and $A_α$ spectral radius where $i\in\{1,2,\lfloor\frac{n}{2}\rfloor\,\lceil\frac{n}{2}\rceil+1,n-3,n-2,n-1\}$, there are some results have been given by Xu, Li and Sun et al., respectively. In 2021, Luo and Guo [Discrete Math., 345 (2022) 112778] determined all graphs in $\mathcal{G}_{n,n-4}$ having the minimum spectral radius. In this paper, we characterize the graphs in $\mathcal{G}_{n,n-4}$ having the minimum and the maximum $A_α$ spectral radius for $α\in[\frac{1}{2},1)$, respectively.

math.CO↗

arXiv2022

Signless Laplacian Estrada index and Laplacian Estrada index of uniform hypergraphs

We generalize the notions of Laplacian and signless Laplacian Estrada index to uniform hypergraphs. For an $r$-uniform hypergraph $H,$ we derive an order $r+1$ trace formula of the (signless) Laplacian tensor of $H.$ Among others by using this trace formula, we obtain lower bounds for the signless Laplacian Estrada index and upper bounds for the Laplacian Estrada index. Moreover, we establish a bound involving both the Laplacian Estrada index and Laplacian energy of a uniform hypergraph.

math.CO↗

arXiv2022

Integer colorings with no rainbow 3-term arithmetic progression

In this paper, we study the rainbow Erdős-Rothschild problem with respect to 3-term arithmetic progressions. We obtain the asymptotic number of $r$-colorings of $[n]$ without rainbow 3-term arithmetic progressions, and we show that the typical colorings with this property are 2-colorings. We also prove that $[n]$ attains the maximum number of rainbow 3-term arithmetic progression-free $r$-colorings among all subsets of $[n]$. Moreover, the exact number of rainbow 3-term arithmetic progression-free $r$-colorings of $\mathbb{Z}_p$ is obtained, where $p$ is any prime and $\mathbb{Z}_p$ is the cyclic group of order $p$.

math.CO↗

arXiv2021

On a conjecture of Nikiforov involving a spectral radius condition for a graph to contain all trees

We partly confirm a Brualdi-Solheid-Turán type conjecture due to Nikiforov, which is a spectral radius analogue of the well-known Erdős-Sós Conjecture that any tree of order $t$ is contained in a graph of average degree greater than $t-2$. We confirm Nikiforov's Conjecture for all brooms and for a larger class of spiders. For our proofs we also obtain a new Turán type result which might turn out to be of independent interest.

math.CO↗

arXiv2021

Spectral radius conditions for the existence of all subtrees of diameter at most four

Let $μ(G)$ denote the spectral radius of a graph $G$. We partly confirm a conjecture due to Nikiforov, which is a spectral radius analogue of the well-known Erdős-Sós Conjecture that any tree of order $t$ is contained in a graph of average degree greater than $t-2$. Let $S_{n,k}=K_{k}\vee\overline{K_{n-k}}$, and let $S_{n,k}^{+}$ be the graph obtained from $S_{n,k}$ by adding a single edge joining two vertices of the independent set of $S_{n,k}$. In 2010, Nikiforov conjectured that for a given integer $k$, every graph $G$ of sufficiently large order $n$ with $μ(G)\geq μ(S_{n,k}^{+})$ contains all trees of order $2k+3$, unless $G=S_{n,k}^{+}$. We confirm this conjecture for trees with diameter at most four, with one exception. In fact, we prove the following stronger result for $k\geq 8$. If a graph $G$ with sufficiently large order $n$ satisfies $μ(G)\geq μ(S_{n,k})$ and $G\neq S_{n,k}$, then $G$ contains all trees of order $2k+3$ with diameter at most four, except for the tree obtained from a star $K_{1,k+1}$ by subdividing each of its $k+1$ edges once.

math.CO↗

arXiv2021

On the $A_α$ spectral radius and $A_α$ energy of non-strongly connected digraphs

Let $A_α(G)$ be the $A_α$-matrix of a digraph $G$ and $λ_{α1}, λ_{α2}, \ldots, λ_{αn}$ be the eigenvalues of $A_α(G)$. Let $ρ_α(G)$ be the $A_α$ spectral radius of $G$ and $E_α(G)=\sum_{i=1}^n λ_{αi}^2$ be the $A_α$ energy of $G$ by using second spectral moment. Let $\mathcal{G}_n^m$ be the set of non-strongly connected digraphs with order $n$, which contain a unique strong component with order $m$ and some directed trees which are hung on each vertex of the strong component. In this paper, we characterize the digraph which has the maximal $A_α$ spectral radius and the maximal (minimal) $A_α$ energy in $\mathcal{G}_n^m$.

math.CO↗

arXiv2021

The Erdős-Gyárfás function with respect to Gallai-colorings

For fixed $p$ and $q$, an edge-coloring of the complete graph $K_n$ is said to be a $(p, q)$-coloring if every $K_p$ receives at least $q$ distinct colors. The function $f(n, p, q)$ is the minimum number of colors needed for $K_n$ to have a $(p, q)$-coloring. This function was introduced about 45 years ago, but was studied systematically by Erdős and Gyárfás in 1997, and is now known as the Erdős-Gyárfás function. In this paper, we study $f(n, p, q)$ with respect to Gallai-colorings, where a Gallai-coloring is an edge-coloring of $K_n$ without rainbow triangles. Combining the two concepts, we consider the function $g(n, p, q)$ that is the minimum number of colors needed for a Gallai-$(p, q)$-coloring of $K_n$. Using the anti-Ramsey number for $K_3$, we have that $g(n, p, q)$ is nontrivial only for $2\leq q\leq p-1$. We give a general lower bound for this function and we study how this function falls off from being equal to $n-1$ when $q=p-1$ and $p\geq 4$ to being $Θ(\log n)$ when $q = 2$. In particular, for appropriate $p$ and $n$, we prove that $g=n-c$ when $q=p-c$ and $c\in \{1,2\}$, $g$ is at most a fractional power of $n$ when $q=\lfloor\sqrt{p-1}\rfloor$, and $g$ is logarithmic in $n$ when $2\leq q\leq \lfloor\log_2 (p-1)\rfloor+1$.

math.CO↗

arXiv2021

Extremal problems and results related to Gallai-colorings

A Gallai-coloring (Gallai-$k$-coloring) is an edge-coloring (with colors from $\{1, 2, \ldots, k\}$) of a complete graph without rainbow triangles. Given a graph $H$ and a positive integer $k$, the $k$-colored Gallai-Ramsey number $GR_k(H)$ is the minimum integer $n$ such that every Gallai-$k$-coloring of the complete graph $K_n$ contains a monochromatic copy of $H$. In this paper, we consider two extremal problems related to Gallai-$k$-colorings. First, we determine upper and lower bounds for the maximum number of edges that are not contained in any rainbow triangle or monochromatic triangle in a $k$-edge-coloring of $K_n$. Second, for $n\geq GR_k(K_3)$, we determine upper and lower bounds for the minimum number of monochromatic triangles in a Gallai-$k$-coloring of $K_{n}$, yielding the exact value for $k=3$. Furthermore, we determine the Gallai-Ramsey number $GR_k(K_4+e)$ for the graph on five vertices consisting of a $K_4$ with a pendant edge.

math.CO↗

arXiv2021

On the $A_α$ spectral radius of strongly connected digraphs

Let $G$ be a digraph with adjacency matrix $A(G)$. Let $D(G)$ be the diagonal matrix with outdegrees of vertices of $G$. Nikiforov \cite{Niki} proposed to study the convex combinations of the adjacency matrix and diagonal matrix of the degrees of undirected graphs. Liu et al. \cite{LWCL} extended the definition to digraphs. For any real $α\in[0,1]$, the matrix $A_α(G)$ of a digraph $G$ is defined 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 $G$, denoted by $λ_α(G)$. This paper proves some extremal results about the spectral radius $λ_α(G)$ that generalize previous results about $λ_0(G)$ and $λ_{\frac{1}{2}}(G)$. In particular, we characterize the extremal digraph with the maximum (or minimum) $A_α$ spectral radius among all $\widetilde{\infty}$-digraphs and $\widetildeθ$-digraphs on $n$ vertices. Furthermore, we determine the digraphs with the second and the third minimum $A_α$ spectral radius among all strongly connected bicyclic digraphs. For $0\leqα\leq\frac{1}{2}$, we also determine the digraphs with the second, the third and the fourth minimum $A_α$ spectral radius among all strongly connected digraphs on $n$ vertices. Finally, we characterize the digraph with the minimum $A_α$ spectral radius among all strongly connected bipartite digraphs which contain a complete bipartite subdigraph.

math.CO↗

arXiv2021

The generalized Turán number of spanning linear forests

Let $\mathcal{F}$ be a family of graphs. A graph $G$ is called \textit{$\mathcal{F}$-free} if for any $F\in \mathcal{F}$, there is no subgraph of $G$ isomorphic to $F$. Given a graph $T$ and a family of graphs $\mathcal{F}$, the generalized Turán number of $\mathcal{F}$ is the maximum number of copies of $T$ in an $\mathcal{F}$-free graph on $n$ vertices, denoted by $ex(n,T,\mathcal{F})$. A linear forest is a graph whose connected components are all paths or isolated vertices. Let $\mathcal{L}_{n,k}$ be the family of all linear forests of order $n$ with $k$ edges and $K^*_{s,t}$ a graph obtained from $K_{s,t}$ by substituting the part of size $s$ with a clique of the same size. In this paper, we determine the exact values of $ex(n,K_s,\mathcal{L}_{n,k})$ and $ex(n,K^*_{s,t},\mathcal{L}_{n,k})$. Also, we study the case of this problem when the \textit{"host graph"} is bipartite. Denote by $ex_{bip}(n,T,\mathcal{F})$ the maximum possible number of copies of $T$ in an $\mathcal{F}$-free bipartite graph with each part of size $n$. We determine the exact value of $ex_{bip}(n,K_{s,t},\mathcal{L}_{n,k})$. Our proof is mainly based on the shifting method.

math.CO↗

arXiv2020

The bounds of the spectral radius of general hypergraphs in terms of clique number

The spectral radius (or the signless Laplacian spectral radius) of a general hypergraph is the maximum modulus of the eigenvalues of its adjacency (or its signless Laplacian) tensor. In this paper, we firstly obtain a lower bound of the spectral radius (or the signless Laplacian spectral radius) of general hypergraphs in terms of clique number. Moreover, we present a relation between a homogeneous polynomial and the clique number of general hypergraphs. As an application, we finally obtain an upper bound of the spectral radius of general hypergraphs in terms of clique number.

math.CO↗

arXiv2020

Iota energy orderings of bicyclic signed digraphs

The concept of energy of a signed digraph is extended to iota energy of a signed digraph. The energy of a signed digraph $S$ is defined by $E(S)=\sum_{k=1}^n|\text{Re}(z_k)|$, where $\text{Re}(z_k)$ is the real part of eigenvalue $z_k$ and $z_k$ is the eigenvalue of the adjacency matrix of $S$ with $n$ vertices, $k=1,2,\ldots,n$. Then the iota energy of $S$ is defined by $E(S)=\sum_{k=1}^n|\text{Im}(z_k)|$, where $\text{Im}(z_k)$ is the imaginary part of eigenvalue $z_k$. In this paper, we consider a special graph class for bicyclic signed digraphs $\mathcal{S}_n$ with $n$ vertices which have two vertex-disjoint signed directed even cycles. We give two iota energy orderings of bicyclic signed digraphs, one is including two positive or two negative directed even cycles, the other is including one positive and one negative directed even cycles.

math.CO↗

arXiv2020

On the Capacity of MIMO Optical Wireless Channels

This paper studies the capacity of a general multiple-input multiple-output (MIMO) free-space optical intensity channel under a per-input-antenna peak-power constraint and a total average-power constraint over all input antennas. The focus is on the scenario with more transmit than receive antennas. In this scenario, different input vectors can yield identical distributions at the output, when they result in the same image vector under multiplication by the channel matrix. We first determine the most energy-efficient input vectors that attain each of these image vectors. Based on this, we derive an equivalent capacity expression in terms of the image vector, and establish new lower and upper bounds on the capacity of this channel. The bounds match when the signal-to-noise ratio (SNR) tends to infinity, establishing the high-SNR asymptotic capacity. We also characterize the low-SNR slope of the capacity of this channel.

cs.IT↗

arXiv2020

Hypothesis Testing Over the Two-hop Relay Network

Coding and testing schemes and the corresponding achievable type-II error exponents are presented for binary hypothesis testing over two-hop relay networks. The schemes are based on cascade source coding techniques and {unanimous decision-forwarding}, the latter meaning that a terminal decides on the null hypothesis only if all previous terminals have decided on the null hypothesis. If the observations at the transmitter, the relay, and the receiver form a Markov chain in this order, then, without loss in performance, the proposed cascade source code can be replaced by two independent point-to-point source codes, one for each hop. The decoupled scheme (combined with decision-forwarding) is shown to attain the optimal type-II error exponents for various instances of "testing against conditional independence." The same decoupling is shown to be optimal also for some instances of "testing against independence," when the observations at the transmitter, the receiver, and the relay form a Markov chain in this order, and when the relay-to-receiver link is of sufficiently high rate. For completeness, the paper also presents an analysis of the Shimokawa-Han-Amari binning scheme for the point-to-point hypothesis testing setup.

cs.IT↗

arXiv2019

Forbidden rainbow subgraphs that force large monochromatic or multicolored k-connected subgraphs

Let $n, k, m$ be positive integers with $n\gg m\gg k$, and let $\mathcal{A}$ be the set of graphs $G$ of order at least 3 such that there is a $k$-connected monochromatic subgraph of order at least $n-f(G,k,m)$ in any rainbow $G$-free coloring of $K_n$ using all the $m$ colors. In this paper, we prove that the set $\mathcal{A}$ consists of precisely $P_6$, $P_3\cup P_4$, $K_2\cup P_5$, $K_2\cup 2P_3$, $2K_2\cup K_3$, $2K_2\cup P^{+}_4$, $3K_2\cup K_{1,3}$ and their subgraphs of order at least 3. Moreover, we show that for any graph $H\in \mathcal{A}$, if $n$ sufficiently larger than $m$ and $k$, then any rainbow $(P_3\cup H)$-free coloring of $K_n$ using all the $m$ colors contains a $k$-connected monochromatic subgraph of order at least $cn$, where $c=c(H)$ is a constant, not depending on $n$, $m$ or $k$. Furthermore, we consider a parallel problem in complete bipartite graphs. Let $s, t, k, m$ be positive integers with ${\rm min}\left\{s, t\right\}\gg m\gg k$ and $m\geq |E(H)|$, and let $\mathcal{B}$ be the set of bipartite graphs $H$ of order at least 3 such that there is a $k$-connected monochromatic subgraph of order at least $s+t-f(H,k,m)$ in any rainbow $H$-free coloring of $K_{s,t}$ using all the $m$ colors, where $f(H,k,m)$ is not depending on $s$ or $t$. We prove that the set $\mathcal{B}$ consists of precisely $2P_3$, $2K_2\cup K_{1,3}$ and their subgraphs of order at least 3. Finally, we consider the large $k$-connected multicolored subgraph instead of monochromatic subgraph. We show that for $1\leq k \leq 3$ and $n$ sufficiently large, every Gallai-3-coloring of $K_n$ contains a $k$-connected subgraph of order at least $n-\left\lfloor\frac{k-1}{2}\right\rfloor$ using at most two colors. We also show that the above statement is false for $k=4t$, where $t$ is an positive integer.

math.CO↗

arXiv2019

Gallai-Ramsey numbers for rainbow paths

Given graphs $G$ and $H$ and a positive integer $k$, the \emph{Gallai-Ramsey number}, denoted by $gr_{k}(G : H)$ is defined to be the minimum integer $n$ such that every coloring of $K_{n}$ using at most $k$ colors will contain either a rainbow copy of $G$ or a monochromatic copy of $H$. We consider this question in the cases where $G \in \{P_{4}, P_{5}\}$. In the case where $G = P_{4}$, we completely solve the Gallai-Ramsey question by reducing to the $2$-color Ramsey numbers. In the case where $G = P_{5}$, we conjecture that the problem reduces to the $3$-color Ramsey numbers and provide several results in support of this conjecture.

math.CO↗

arXiv2018

The largest signless Laplacian spectral radius of uniform supertrees with diameter and pendent edges (vertices)

Let $S_{1}(m, d, k)$ be the $k$-uniform supertree obtained from a loose path $P:v_{1}, e_{1}, v_{2}, \ldots,v_{d}, e_{d}, v_{d+1}$ with length $d$ by attaching $m-d$ edges at vertex $v_{\lfloor\frac{d}{2}\rfloor+1}.$ Let $\mathbb{S}(m,d,k)$ be the set of $k$-uniform supertrees with $m$ edges and diameter $d$ and $q(G)$ be the signless Laplacian spectral radius of a $k$-uniform hypergraph $G$. In this paper, we mainly determine $S_{1}(m,d,k)$ with the largest signless Laplacian spectral radius among all supertrees in $\mathbb{S}(m,d,k)$ for $3\leq d\leq m-1$. Furthermore, we determine the unique uniform supertree with the maximum signless Laplacian spectral radius among all the uniform supertrees with $n$ vertices and pendent edges (vertices).

math.CO↗

arXiv2018

The generalized distance matrix of digraphs

Let $D(G)$ and $D^Q(G)= Diag(Tr) + D(G)$ be the distance matrix and distance signless Laplacian matrix of a simple strongly connected digraph $G$, respectively, where $Diag(Tr)=\textrm{diag}(D_1,D_2,$ $\ldots,D_n)$ be the diagonal matrix with vertex transmissions of the digraph $G$. To track the gradual change of $D(G)$ into $D^Q(G)$, in this paper, we propose to study the convex combinations of $D(G)$ and $Diag(Tr)$ defined by $$D_α(G)=αDiag(Tr)+(1-α)D(G), \ \ 0\leq α\leq1.$$ This study reduces to merging the distance spectral and distance signless Laplacian spectral theories. The eigenvalue with the largest modulus of $D_α(G)$ is called the $D_α$ spectral radius of $G$, denoted by $μ_α(G)$. We determine the digraph which attains the maximum (or minimum) $D_α$ spectral radius among all strongly connected digraphs. Moreover, we also determine the digraphs which attain the minimum $D_α$ spectral radius among all strongly connected digraphs with given parameters such as dichromatic number, vertex connectivity or arc connectivity.

math.CO↗