arXiv subjects
Publications and source records attributed to Ligong Wang.
A graph $G$ is called $k$-factor-critical if $G-S$ has a perfect matching for every $S\subseteq G$ with $|S|=k$. A connected graph $G$ is called $t$-connected if it has more than $t$ vertices and remains connected whenever fewer than $t$ vertices are removed. We give a condition on the number of edges and a condition on the spectral radius for $k$-factor-criticality in $t$-connected graphs.
Consider the problem of distributed binary hypothesis testing with two terminals, where the decision is made at one of them (the "receiver"). We study the exponent of the error probability of the second type. Previously, an achievable exponent was derived by Shimokawa, Han, and Amari using a "quantization and binning" scheme. We propose a simple modification on the receiver's decision rule in this scheme to attain a better exponent.
Let $G$ be a strongly connected digraph with $n$ vertices and $m$ arcs. For any real $α\in[0,1]$, the $A_α$ matrix of a digraph $G$ is defined as $$A_α(G)=αD(G)+(1-α)A(G),$$ where $A(G)$ is the adjacency matrix of $G$ and $D(G)$ is the outdegrees diagonal matrix of $G$. The eigenvalue of $A_α(G)$ with the largest modulus is called the $A_α$ spectral radius of $G$, denoted by $λ_α(G)$. In this paper, we first obtain an upper bound on $λ_α(G)$ for $α\in[\frac{1}{2},1)$. Employing this upper bound, we prove that for two strongly connected digraphs $G_1$ and $G_2$ with $n\ge4$ vertices and $m$ arcs, and $α\in [\frac{1}{\sqrt{2}},1)$, if the maximum outdegree $Δ^+(G_1)\ge 2α(1-α)(m-n+1)+2α$ and $Δ^+(G_1)>Δ^+(G_2)$, then $λ_α(G_1)>λ_α(G_2)$. Moreover, We also give another upper bound on $λ_α(G)$ for $α\in[\frac{1}{2},1)$. Employing this upper bound, we prove that for two strongly connected digraphs with $m$ arcs, and $α\in[\frac{1}{2},1)$, if the maximum outdegree $Δ^+(G_1)>\frac{2m}{3}+1$ and $Δ^+(G_1)>Δ^+(G_2)$, then $λ_α(G_1)+\frac{1}{4}>λ_α(G_2)$.
We study the fundamental limits of covert communications over general memoryless additive-noise channels. We assume that the legitimate receiver and the eavesdropper share the same channel and therefore see the same outputs. Under mild integrability assumptions, we find a general upper bound on the square-root scaling constant, which only involves the variance of the logarithm of the probability density function of the noise. Furthermore, we show that, under some additional assumptions, this upper bound is tight. We also provide upper bounds on the length of the secret key required to achieve the optimal scaling.
Let $Q(G)=D(G)+A(G)$ be the signless Laplacian matrix of a simple graph $G$, where $D(G)$ and $A(G)$ are the degree diagonal matrix and the adjacency matrix of $G$, respectively. The largest eigenvalue of $Q(G)$, denoted by $q(G)$, is called the signless Laplacian spectral radius of $G$. Let $θ(l_{1},l_{2},l_{3})$ denote the theta graph which consists of two vertices connected by three internally disjoint paths with length $l_{1}$, $l_{2}$ and $l_{3}$. Let $F_{n}$ be the friendship graph consisting of $\frac{n-1}{2}$ triangles which intersect in exactly one common vertex for odd $n\geq3$ and obtained by hanging an edge to the center of $F_{n-1}$ for even $n\geq4$. Let $S_{n,k}$ denote the graph obtained by joining each vertex of $K_{k}$ to $n-k$ isolated vertices. Let $S_{n,k}^{+}$ denote the graph obtained by adding an edge to the two isolated vertices of $S_{n,k}$. In this paper, firstly, we show that if $G$ is $θ(1,2,2)$-free, then $q(G)\leq q(F_{n})$, unless $G\cong F_{n}$. Secondly, we show that if $G$ is $θ(1,2,3)$-free, then $q(G)\leq q(S_{n,2})$, unless $G\cong S_{n,2}$. Finally, we show that if $G$ is $\{θ(1,2,2),F_{5}\}$-free, then $q(G)\leq q(S_{n,1}^{+})$, unless $G\cong S_{n,1}^{+}$.
A graph $G$ is $F$-free if $G$ does not contain $F$ as a subgraph. Let $\mathcal{G}(m, F)$ denote the family of $F$-free graphs with $m$ edges and without isolated vertices. Let $S_{n,k}$ denote the graph obtained by joining every vertex of $K_{k}$ to $n-k$ isolated vertices and $S_{n,k}^{t}$ denote the graph obtained from $S_{n-t,k}$ by attaching $t$ pendant vertices to the maximal degree vertex of $S_{n-t,k}$, respectively. Denote by $H_{n}$ the fan graph obtain from $n-1$-vertex path plus a vertex adjacent to each vertex of the path. Particularly, the graph $H_{5}$ is also known as the gem. Zhang and Wang [Discrete Math. 347(2024)114171] and Yu, Li and Peng [arXiv: 2404. 03423] showed that every gem-free graph $G$ with $m$ edges satisfies $ρ(G)\leq ρ(S_{\frac{m+3}{2},2})$. In this paper, we show that if $G\in \mathcal{G}(m, H_{5})\setminus S_{\frac{m+3}{2},2}$ be a graph of odd size $m\geq23$, then $ρ(G)\leq ρ(S_{\frac{m+5}{2},2}^{2})$, and equality holds if and only if $G\cong S_{\frac{m+5}{2},2}^{2}$.
A graph $G$ is $F$-free if $G$ does not contain $F$ as a subgraph. Let $ρ(G)$ be the spectral radius of a graph $G$. Let $θ(1,p,q)$ denote the theta graph, which is obtained by connecting two distinct vertices with three internally disjoint paths with lengths $1, p, q$, where $p\leq q$. Let $S_{n,k}$ denote the graph obtained by joining every vertex of $K_{k}$ to $n-k$ isolated vertices and $S_{n,k}^{-}$ denote the graph obtained from $S_{n,k}$ by deleting an edge incident to a vertex of degree $k$, respectively. In this paper, we show that if $ρ(G)\geqρ(S_{\frac{m+4}{2},2}^{-})$ for a graph $G$ with even size $m\geq 92$, then $G$ contains a $θ(1,3,3)$ unless $G\cong S_{\frac{m+4}{2},2}^{-}$.
The $Q$-index of graph $G$ is the largest eigenvalue of the signless Laplacian matrix of $G$. Wang [Discrete Appl. Math. 356(2024)] proved the sharp upper bounds on the $Q$-index of leaf-free graphs with given size and characterized the corresponding extremal graphs. A graph is $2$ leaves-free if it has no two pendent vertices. In this paper, we give sharp upper bounds on the $Q$-index of 2 leaves-free graphs with given size and characterize the corresponding extremal graphs.
Let $T_{4p}=\langle a,b\mid a^{2p}=1,a^p=b^2, b^{-1}ab=a^{-1}\rangle$ be the dicyclic group of order $4p$. A Cayley digraph over $T_{4p}$ is called a dicirculant digraph. In this paper, we calculate the number of (connected) dicirculant digraphs of order $4p$ ($p$ prime) up to isomorphism by using the Pólya Enumeration Theorem. Moreover, we get the number of (connected) dicirculant digraphs of order $4p$ ($p$ prime) and out-degree $k$ for every $k$.
Let $\mathcal{F}$ be a family of $r$-uniform hypergraphs. Denote by $\ex^{\mathrm{conn}}_r(n,\mathcal{F})$ the maximum number of hyperedges in an $n$-vertex connected $r$-uniform hypergraph which contains no member of $\mathcal{F}$ as a subhypergraph. Denote by $\mathcal{B}C_k$ the Berge cycle of length $k$, and by $\mathcal{B}P_k$ the Berge path of length $k$. Füredi, Kostochka and Luo, and independently Győri, Salia and Zamora determined $\ex^{\mathrm{conn}}_r(n,\mathcal{B}P_k)$ provided $k$ is large enough compared to $r$ and $n$ is sufficiently large. For the case $k\le r$, Kostochka and Luo obtained an upper bound for $\ex^{\mathrm{conn}}_r(n,\mathcal{B}P_k)$. In this paper, we continue investigating the case $k\le r$. We precisely determine $\ex^{\mathrm{conn}}_r(n,\mathcal{B}P_k)$ when $n$ is sufficiently large and $n$ is not a multiple of~$r$. For the case $k=r+1$, we determine $\ex^{\mathrm{conn}}_r(n,\mathcal{B}P_k)$ asymptotically.
Given two graphs $G, H$ and a positive integer $q$, an $(H,q)$-coloring of $G$ is an edge-coloring of $G$ such that every copy of $H$ in $G$ receives at least $q$ distinct colors. The bipartite Erdős-Gyárfás function $r(K_{n,n}, K_{s,t}, q)$ is defined to be the minimum number of colors needed for $K_{n,n}$ to have a $(K_{s,t}, q)$-coloring. For balanced complete bipartite graphs $K_{p,p}$, the function $r(K_{n,n}, K_{p,p}, q)$ was studied systematically in [Axenovich, Füredi and Mubayi, {\it J. Combin. Theory Ser. B} {\bf 79} (2000), 66--86]. In this paper, we study the asymptotic behavior of this function for complete bipartite graphs $K_{s,t}$ that are not necessarily balanced. Our main results deal with thresholds and lower and upper bounds for the growth rate of this function, in particular for (sub)linear and (sub)quadratic growth. We also obtain new lower bounds for the balanced bipartite case, and improve several results given by Axenovich, Füredi and Mubayi. Our proof techniques are based on an extension to bipartite graphs of the recently developed Color Energy Method by Pohoata and Sheffer and its refinements, and a generalization of an old result due to Corrádi.
In this paper, we study dual quaternion and dual complex unit gain graphs and their spectral properties in a unified frame of dual unit gain graphs. Unit dual quaternions represent rigid movements in the 3D space, and have wide applications in robotics and computer graphics. Dual complex numbers found application in brain science recently. We establish the interlacing theorem for dual unit gain graphs, and show that the spectral radius of a dual unit gain graph is always not greater than the spectral radius of the underlying graph, and these two radii are equal if and only if the dual gain graph is balanced. By using the dual cosine functions, we establish the closed form of eigenvalues of adjacency and Laplacian matrices of dual complex and quaternion unit gain cycles. We then show the coefficient theorem holds for dual unit gain graphs. Similar results hold for the spectral radius of the Laplacian matrix of the dual unit gain graph too.
Let $H_7$ denote the $7$-vertex \textit{fan graph} consisting of a $6$-vertex path plus a vertex adjacent to each vertex of the path. Let $K_3 \vee \frac{m-3}{3}K_1$ be the graph obtained by joining each vertex of a triangle $K_3$ to $\frac{m-3}{3}$ isolated vertices. In this paper, we show that if $G$ is an $H_{7}$-free graph with size $m\geq 33$, then the spectral radius $ ρ(G)\leq 1+\sqrt{m-2},$ equality holds if and only if $G\cong K_3 \vee \frac{m-3}{3}K_1$ (possibly, with some isolated vertices).
Let $G=(V(G), E(G))$ be a graph with vertex set $V(G)$ and edge set $E(G)$. For $k\geq2$ and given any subset $S\subseteq|V(G)|$ with $|S|=k$, if a graph $G$ of order $|V(G)|\geq k+1$ always has a spanning tree $T$ such that $S$ is precisely the set of leaves of $T$, then the graph $G$ is a $k$-leaf-connected graph. A graph $G$ is called Hamilton-connected if any two vertices of $G$ are connected by a Hamilton path. Based on the definitions of $k$-leaf-connected and Hamilton-connected, we known that a graph is $2$-leaf-connected if and only if it is Hamilton-connected. During the past decades, there have been many results of sufficient conditions for Hamilton-connected with respect to topological indices. In this paper, we present sufficient conditions for a graph $G$ to be $k$-leaf-connected in terms of the Zagreb index, the reciprocal degree distance or the hyper-Zagreb index. Furthermore, we use the first Zagreb index and hyper-Zagreb index of the complement graph $\overline{G}$ to give sufficient conditions for a graph $G$ to be $k$-leaf-connected.
We consider the wiretap channel, where the individual channel uses have memory or are influenced by an adversary. We analyze the explicit and computationally efficient construction of information-theoretically secure coding schemes which use the inverse of an extractor and an error-correcting code. These schemes are known to achieve secrecy capacity on a large class of memoryless wiretap channels. We show that this also holds for certain channel types with memory. In particular, they can achieve secrecy capacity on channels where an adversary can pick a sequence of ``states'' governing the channel's behavior, as long as, given every possible state, the channel is strongly symmetric.
An $r$-graph $H$ is a hypergraph consisting of a nonempty set of vertices $V$ and a collection of $r$-element subsets of $V$ we refer to as the edges of $H$. An $r$-graph $H$ is called linear if any two edges of $H$ intersect in at most one vertex. Let $F$ and $H$ be two linear $r$-graphs. If $H$ contains no copy of $F$, then $H$ is called $F$-free. The linear Turán number of $F$, denoted by $ex_r^{lin}(n,F)$, is the maximum number of edges in any $F$-free $n$-vertex linear $r$-graph. The crown $C_{13}$ (or $E_4$) is a linear 3-graph which is obtained from three pairwise disjoint edges by adding one edge that intersects all three of them in one vertex. In 2022, Gyárfás, Ruszinkó and Sárközy initiated the study of $ex_3^{lin}(n,F)$ for different choices of an acyclic 3-graph $F$. They determined the linear Turán numbers for all linear 3-graphs with at most 4 edges, except the crown. They established lower and upper bounds for $ex_3^{lin}(n,C_{13})$. In fact, their lower bound on $ex_3^{lin}(n,C_{13})$ is essentially tight, as was shown in a recent paper by Tang, Wu, Zhang and Zheng. In this paper, we generalize the notion of a crown to linear $r$-graphs for $r\ge 3$, and also generalize the above results to linear $r$-graphs.
In this paper, we investigate the existence of fractional revival on Cayley graphs over finite abelian groups. We give a necessary and sufficient condition for Cayley graphs over finite abelian groups to have fractional revival. As applications, the existence of fractional revival on circulant graphs and cubelike graphs are characterized.
In this paper, we study the integral Cayley graphs over a non-abelian group $U_{6n}=\langle a,b\mid a^{2n}=b^3=1, a^{-1}ba=b^{-1}\rangle$ of order $6n$. We give a necessary and sufficient condition for the integrality of Cayley graphs over $U_{6n}$. We also study relationships between the integrality of Cayley graphs over $U_{6n}$ and the Boolean algebra of cyclic groups. As applications, we construct some infinite families of connected integral Cayley graphs over $U_{6n}$.