arXiv subjects

Ligong Wang

Publications and source records attributed to Ligong Wang.

At least 55 records · Page 3Linked to original sources

arXiv2024

DQ-integral and DL-integral generalized wheel graphs

A graph G is said to be M-integral (resp. A-integral, D-integral, DL-integral or DQ-integral) if all eigenvalues of its matrix M (resp. adjacency matrix A(G), distance matrix D(G), distance Laplacian matrix DL(G) or distance signless Laplacian matrix DQ(G)) are integers. Lu et al. [Discrete Math, 346 (2023)] defined the generalized wheel graph GW(a, m, n) as the join of two regular graphs aKm and Cn, and obtained all D-integral generalized wheel graphs. Based on the above research, in this paper, we determine all DL-integral and DQ-integral generalized wheel graphs respectively. As byproducts, we give a sufficient and necessary condition for the join of two regular graphs G1 and G2 to be DL-integral, from which we can get infinitely many new classes of DL-integral graphs according to the large number of research results about the A-integral graphs.

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arXiv2023

Pretty good fractional revival on Cayley graphs over dicyclic groups

In this paper, we investigate the existence of pretty good fractional revival on Cayley graphs over dicyclic groups. We first give a necessary and sufficient description for Cayley graphs over dicyclic groups admitting pretty good fractional revival. By this description, we give some sufficient conditions for Cayley graphs over dicyclic groups admitting or not admitting pretty good fractional revival.

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arXiv2023

State-Dependent Channels with a Message-Cognizant Helper

The capacity of a state-dependent discrete memoryless channel (SD-DMC) is derived for the setting where a message-cognizant rate-limited helper observes the state sequence noncausally, produces its description, and provides the description to both encoder and decoder.

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arXiv2023

Distance spectral conditions for $ID$-factor-critical and fractional $[a, b]$-factor of graphs

Let $G=(V(G), E(G))$ be a graph with vertex set $V(G)$ and edge set $E(G)$. A graph is $ID$-factor-critical if for every independent set $I$ of $G$ whose size has the same parity as $|V(G)|$, $G-I$ has a perfect matching. For two positive integers $a$ and $b$ with $a\leq b$, let $h$: $E(G)\rightarrow [0, 1]$ be a function on $E(G)$ satisfying $a\leq\sum _{e\in E_{G}(v_{i})}h(e)\leq b$ for any vertex $v_{i}\in V(G)$. Then the spanning subgraph with edge set $E_{h}$, denoted by $G[E_{h}]$, is called a fractional $[a, b]$-factor of $G$ with indicator function $h$, where $E_{h}=\{e\in E(G)\mid h(e)>0\}$ and $E_{G}(v_{i})=\{e\in E(G)\mid e$ is incident with $v_{i}$ in $G$\}. A graph is defined as a fractional $[a, b]$-deleted graph if for any $e\in E(G)$, $G-e$ contains a fractional $[a, b]$-factor. For any integer $k\geq 1$, a graph has a $k$-factor if it contains a $k$-regular spanning subgraph. In this paper, we firstly give a distance spectral radius condition of $G$ to guarantee that $G$ is $ID$-factor-critical. Furthermore, we provide sufficient conditions in terms of distance spectral radius and distance signless Laplacian spectral radius for a graph to contain a fractional $[a, b]$-factor, fractional $[a, b]$-deleted-factor and $k$-factor.

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arXiv2023

Message-Cognizant Assistance and Feedback for the Gaussian Channel

A formula is derived for the capacity of the Gaussian channel with a benevolent message-cognizant rate-limited helper that provides a noncausal description of the noise to the encoder and decoder. This capacity is strictly larger than when the helper is message oblivious, with the difference being particularly pronounced at low signal-to-noise ratios. It is shown that in this setup, a feedback link from the receiver to the encoder does not increase capacity. However, in the presence of such a link, said capacity can be achieved even if the helper is oblivious to the transmitted message.

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arXiv2023

The characteristic polynomials of uniform hypercycles with length four

Let $C_{m}$ be a cycle with length $m.$ The $k$-uniform hypercycle with length $m$ obtained by adding $k-2$ new vertices in every edge of $C_{m},$ denoted by $C_{m,k}.$ In this paper, we obtain some trace formulas of uniform hypercycles with length four. Moreover, we give the characteristic polynomials of uniform hypercycles with length four.

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arXiv2023

The $α$-index of graphs without intersecting triangles/quadrangles as a minor

The $A_α$-matrix of a graph $G$ is the convex linear combination of the adjacency matrix $A(G)$ and the diagonal matrix of vertex degrees $D(G)$, i.e., $A_α(G) = αD(G) + (1 - α)A(G)$, where $0\leqα\leq1$. The $α$-index of $G$ is the largest eigenvalue of $A_α(G)$. Particularly, the matrix $A_0(G)$ (resp. $2A_{\frac{1}{2}}(G)$) is exactly the adjacency matrix (resp. signless Laplacian matrix) of $G$. He, Li and Feng [arXiv:2301.06008 (2023)] determined the extremal graphs with maximum adjacency spectral radius among all graphs of sufficiently large order without intersecting triangles and quadrangles as a minor, respectively. Motivated by the above results of He, Li and Feng, in this paper we characterize the extremal graphs with maximum $α$-index among all graphs of sufficiently large order without intersecting triangles and quadrangles as a minor for any $0<α<1$, respectively. As by-products, we determine the extremal graphs with maximum signless Laplacian radius among all graphs of sufficiently large order without intersecting triangles and quadrangles as a minor, respectively.

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arXiv2023

State-Dependent DMC with a Causal Helper

A memoryless state sequence governing the behavior of a memoryless state-dependent channel is to be described causally to an encoder wishing to communicate over said channel. Given the maximal-allowed description rate, we seek the description that maximizes the Shannon capacity. It is shown that the maximum need not be achieved by a memoryless (symbol-by-symbol) description. Such descriptions are, however, optimal when the receiver is cognizant of the state sequence or when the description is allowed to depend on the message. For other cases, a block-Markov scheme with backward decoding is proposed.

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arXiv2023

Fractional revival on semi-Cayley graphs over abelian groups

In this paper, we investigate the existence of fractional revival on semi-Cayley graphs over finite abelian groups. We give some necessary and sufficient conditions for semi-Cayley graphs over finite abelian groups admitting fractional revival. We also show that integrality is necessary for some semi-Cayley graphs admitting fractional revival. Moreover, we characterize the minimum time when semi-Cayley graphs admit fractional revival. As applications, we give examples of certain Cayley graphs over the generalized dihedral groups and generalized dicyclic groups admitting fractional revival.

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arXiv2023

On the $α$-index of minimally $k$-(edge-)connected graphs for small $k$

Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be the diagonal matrix of vertex degrees of $G$. For any real $α\in [0,1]$, Nikiforov defined the $A_α$-matrix of a graph $G$ as $A_α(G)=αD(G)+(1-α)A(G)$. The largest eigenvalue of $A_α(G)$ is called the $α$-index or the $A_α$-spectral radius of $G$. A graph is minimally $k$-(edge)-connected if it is $k$-(edge)-connected and deleting any arbitrary chosen edge always leaves a graph which is not $k$-(edge)-connected. In this paper, we characterize the minimally 2-edge-connected graphs and minimally 3-connected graph with given order having the maximum $α$-index for $α\in [\frac{1}{2},1)$, respectively.

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arXiv2023

Turán numbers of general hypergraph star forests

Let $\mathcal{F}$ be a family of $r$-uniform hypergraphs, and let $H$ be an $r$-uniform hypergraph. Then $H$ is called $\mathcal{F}$-free if it does not contain any member of $\mathcal{F}$ as a subhypergraph. The Turán number of $\mathcal{F}$, denoted by $ex_r(n,\mathcal{F})$, is the maximum number of hyperedges in an $\mathcal{F}$-free $n$-vertex $r$-uniform hypergraph. Our current results are motivated by earlier results on Turán numbers of star forests and hypergraph star forests. In particular, Lidický, Liu and Palmer [Electron. J. Combin. 20 (2013)] determined the Turán number $ex(n,F)$ of a star forest $F$ for sufficiently large $n$. Recently, Khormali and Palmer [European. J. Combin. 102 (2022) 103506] generalized the above result to three different well-studied hypergraph settings, but restricted to the case that all stars in the hypergraph star forests are identical. We further generalize these results to general hypergraph star forests.

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arXiv2023

Sharp bounds for Laplacian spectral moments of digraphs with a fixed dichromatic number

The $k$-th Laplacian spectral moment of a digraph $G$ is defined as $\sum_{i=1}^n λ_i^k$, where $λ_i$ are the eigenvalues of the Laplacian matrix of $G$ and $k$ is a nonnegative integer. For $k=2$, this invariant is better known as the Laplacian energy of $G$. We extend recently published results by characterizing the digraphs which attain the minimal and maximal Laplacian energy within classes of digraphs with a fixed dichromatic number. We also determine sharp bounds for the third Laplacian spectral moment within the special subclass which we define as join digraphs. We leave the full characterization of the extremal digraphs for $k\ge 3$ as an open problem.

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arXiv2023

Covert Communication over Two Types of Additive Noise Channels

We extend previous results on covert communication over the additive white Gaussian noise channel to two other types of additive noise channels. The first is the Gaussian channel with memory, where the noise sequence is a Gaussian vector with an arbitrary invertible covariance matrix. We show that the fundamental limit for covert communication over such a channel is the same as over the channel with white, i.e., memoryless, Gaussian noise. The second type of channel we consider is one with memoryless generalized Gaussian noise. For such a channel we prove a general upper bound on the dominant term in the maximum number of nats that can be covertly communicated over n channel uses. When the shape parameter p of the generalized Gaussian noise distribution is in the interval (0, 1], we also prove a matching lower bound.

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arXiv2023

The $A_α$ spectral radius of $k$-connected graphs with given diameter

Let $G$ be a graph with adjacency matrix $A(G)$ and degree diagonal matrix $D (G)$. In 2017, Nikiforov defined the matrix $A_α(G) = αD(G) + (1-α)A(G)$ for any real $α\in[0,1]$. The largest eigenvalue of $A_α(G)$ is called the $A_α$ spectral radius or the $A_α$-index of $G$. Let $\mathcal{G}_{n,k}^d$ be the set of $k$-connected graphs of order $n$ with diameter $d$. In this paper, we determine the graphs with maximum $A_α$ spectral radius among all graphs in $\mathcal{G}_{n,k}^d$ for any $α\in[0,1)$, where $k\geq2$ and $d\geq2$. We generalizes the results about adjacency matrix of Theorem 3.6 in [P. Huang, W.C. Shiu, P.K. Sun, Linear Algebra Appl., 488 (2016) 350--362] and the results about signless Laplacian matrix of Theorem 3.4 in [P. Huang, J.X. Li, W.C. Shiu, Linear Algebra Appl., 617 (2021) 78--99]. Furthermore, we also obtain the upper and lower bounds of the extremal graph in $\mathcal{G}_{n,k}^d$.

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arXiv2022

Linear Turán numbers of acyclic quadruple systems

A linear $r$-uniform hypergraph is called acycilc if it can be constructed starting from one single edge then at each step adding a new edge that intersect the union of the vertices of the previous edges in at most one vertex. Recently, Gyárfás, Ruszinkó and Sárk\''{o}zy initiated the study of the linear Turán numbers of acyclic linear triple systems. In this paper, we extend their results to linear quadruple systems. Here, we concentrate on small trees, paths and matchings. For the case of small trees, we find that for a linear tree $T$, $ex^{lin}_{4}(n,T)$ relates to difficult problems on Steiner system $S(2,4,n)$ For example, we show that $ex^{lin}_{4}(n, P_4)\le \frac{5n}{4}$ with equality holds if and only if the linear quadruple system is the disjoint union of $S(2,4,16)$. Denote by $E^{+}_{4}$ the linear tree consisting of three pairwise disjoint quadruples and a fourth one intersecting all of them. We prove that $12\lfloor\frac{n-4}{9}\rfloor\le ex^{lin}_{4}(n, E^{+}_4)\le \frac{14(n-s)}{9}$, where $s$ is the number of vertices in $G$ with degree at least 8. Denote by $M_k$ and $P_k$ the set of $k$ pairwise disjoint quadruples and the linear path with $k$ quadruples, respectively. For the case of paths, we show that $ex^{lin}_{4}(n, P_k)\le 2.5kn$. For the case of matchings, we prove that for fixed $k$ and sufficiently large $n$, $ex^{lin}_{4}(n, M_k)=g(n,k)$ where $g(n,k$) denotes the maximum number of quadruples that can intersect $k-1$ vertices in a linear quadruple system on $n$ vertices.

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arXiv2022

An improvement of sufficient condition for $k$-leaf-connected graphs

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.

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arXiv2022

On the $α$-index of minimally 2-connected graphs with given order or size

For any real $α\in [0,1]$, Nikiforov defined the $A_α$-matrix of a graph $G$ as $A_α(G)=αD(G)+(1-α)A(G)$, where $A(G)$ and $D(G)$ are the adjacency matrix and the diagonal matrix of vertex degrees of $G$, respectively. The largest eigenvalue of $A_α(G)$ is called the $α$-index or the $A_α$-spectral radius of $G$. A graph is minimally $k$-connected if it is $k$-connected and deleting any arbitrary chosen edge always leaves a graph which is not $k$-connected. In this paper, we characterize the extremal graphs with the maximum $α$-index for $α\in [\frac{1}{2},1)$ among all minimally 2-connected graphs with given order or size, respectively.

math.CO↗

arXiv2022

Spectral radius of graphs of given size with forbidden subgraphs

Let $ρ(G)$ be the spectral radius of a graph $G$ with $m$ edges. Let $S_{m-k+1}^{k}$ be the graph obtained from $K_{1,m-k}$ by adding $k$ disjoint edges within its independent set. Nosal's theorem states that if $ρ(G)>\sqrt{m}$, then $G$ contains a triangle. Zhai and Shu showed that any non-bipartite graph $G$ with $m\geq26$ and $ρ(G)\geqρ(S_{m}^{1})>\sqrt{m-1}$ contains a quadrilateral unless $G\cong S_{m}^{1}$ [M.Q. Zhai, J.L. Shu, Discrete Math. 345 (2022) 112630]. Wang proved that if $ρ(G)\geq\sqrt{m-1}$ for a graph $G$ with size $m\geq27$, then $G$ contains a quadrilateral unless $G$ is one of four exceptional graphs [Z.W. Wang, Discrete Math. 345 (2022) 112973]. In this paper, we show that any non-bipartite graph $G$ with size $m\geq51$ and $ρ(G)\geqρ(S_{m-1}^{2})>\sqrt{m-2}$ contains a quadrilateral unless $G$ is one of three exceptional graphs. Moreover, we show that if $ρ(G)\geqρ(S_{\frac{m+4}{2},2}^{-})$ for a graph $G$ with even size $m\geq74$, then $G$ contains a $C_{5}^{+}$ unless $G\cong S_{\frac{m+4}{2},2}^{-}$, where $C_{t}^{+}$ denotes the graph obtained from $C_{t}$ and $C_{3}$ by identifying an edge, $S_{n,k}$ denotes the graph obtained by joining each vertex of $K_{k}$ to $n-k$ isolated vertices and $S_{n,k}^{-}$ denotes the graph obtained by deleting an edge incident to a vertex of degree two, respectively.

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