arXiv subjects

Ligong Wang

Publications and source records attributed to Ligong Wang.

At least 19 recordsLinked to original sources

arXiv2026

Some results on the distance spectral radius and edge-disjoint spanning trees of graphs

Let $τ(G)$ denote the maximum number of edge-disjoint spanning trees in a connected graph $G$ of order $n$, and let $ρ_D(G)$ denote its distance spectral radius. For an integer $k\ge2$, Fan, He and Zhao [Discrete Appl. Math. 376 (2025) 31--40] obtained a sharp distance spectral radius condition for $τ(G)\ge k$ when $n\ge2k+6$. In this paper, we fill the gap $2k\le n\le2k+5$ and thus complete the result for all $n\ge2k$. The extremal graph given by Fan, He and Zhao remains valid for $n\ge2k+2$, while we determine the unique extremal graph for each of the orders $n=2k$ and $n=2k+1$. We further obtain sharp distance spectral radius conditions and characterize all extremal graphs under the minimum degree condition $δ(G)\ge k$ for all $n\ge2k$. Finally, for graphs with the stronger minimum degree condition $δ(G)\ge6k-4$ and order $n\ge2δ(G)+2$, we obtain a sharp distance spectral radius condition ensuring $τ(G)\ge k$ and determine the unique extremal graph.

math.CO↗

arXiv2026

Extremal spectral result of outerplanar graphs without $P_{3\cdot l}$

A graph $G$ is $F$-free if it does not contain $F$ as a subgraph. Let $\mathrm{spex}(n,F)$ be the maximum spectral radius over all $n$-vertex $F$-free outerplanar graphs. For integers $t\geq1$ and $l\geq2$, let $P_{t\cdot l}$ be the starlike tree with $t$ branches of length $l-1$. For sufficiently large $n$, Yin, Li, and Meng [arXiv:2504.04364v1] characterized the unique extremal graph for $\mathrm{spex}(n,P_{t\cdot l})$ when $t=1$, $t=2$, or $t\geq4$. They left the case $t=3$ open and proposed a natural candidate for the extremal graph. We show that this candidate is not extremal and determine the unique extremal graph for $\mathrm{spex}(n,P_{3\cdot l})$. For every $l\geq3$ and all sufficiently large $n$, this unique extremal graph is $K_1\vee\bigl(2P_{2l-3}\cup qP_{l-2}\cup P_r\bigr),$ where $q$ and $r$ are integers satisfying $n=2(2l-3)+q(l-2)+r+1,$ $q\geq0,$ $0\leq r<l-2.$

math.CO↗

arXiv2026

Chernoff-Stein-Type Exponent in Testing Between Two Outlier Distributions

Among $2^{nR}$ length-$n$ random sequences, one sequence is an outlier whose index is random. Under hypothesis $\mathcal{H}_0$, the components of the outlier are independent and identically distributed (IID) according to $Q_0$, whereas under hypothesis $\mathcal{H}_1$ they are IID according to $Q_1$. The remaining $(2^{nR}-1)$ sequences are mutually independent, independent of the outlier, and IID according to $P$ under both hypotheses. Based on the observation of all $2^{nR}$ sequences, one wishes to decide between $\mathcal{H}_0$ and $\mathcal{H}_1$. Under the constraint that the decision error probability under $\mathcal{H}_0$ must be bounded away from $1$, we determine the fastest exponential decay rate of the decision error probability under $\mathcal{H}_1$.

cs.IT↗

arXiv2026

Spectral extremal problems for fractional $ID$-$[a,b]$-factor-critical graphs

A factor of a graph is essentially a specific type spanning subgraph. In recent years, the spectral extremal problem of characterizing the existence of graph factors via eigenvalues has been widely studied. This paper focuses on fractional $ID$-$[a, b]$-factor-critical graphs, which are a natural generalization of fractional $[a,b]$-factors. Let $r \ge 1$ be an integer. A graph $G$ is fractional $ID$-$[a, b]$-factor-critical if for every independent set $I$ of $G$ with $|I| = r$, $G - I$ has a fractional $[a, b]$-factor. In 2026, Jia, Fan and Liu posed the spectral version conjecture for a graph to be fractional $ID$-$[a, b]$-factor-critical [Linear Algebra Appl. 732 (2026) 1-17]. In this paper, we first prove the conjecture holds for connected graphs when $b\ge 2r+2$. Furthermore, for minimum degree $δ(G)\ge a+r$, we present spectral radius and size conditions that ensure a graph is fractional $ID$-$[a, b]$-factor-critical, which improve the results of Jia, Fan and Liu.

math.CO↗

arXiv2026

Spectral radius conditions for edge-disjoint spanning trees in $(k+c)$-edge-connected graphs

Let $τ(G)$ denote the spanning tree packing number of a graph $G$. Recently, Zhang and Fan [J. Graph Theory 112 (2) (2026) 128--144] posed the problem of finding a tight spectral radius condition for an $m$-edge-connected graph $G$ to guarantee $τ(G)\ge k$ for $k+1\le m\le 2k-1$. They solved the cases $m=k$ and $k=2, m=3$. In this paper, we study this problem for all $m=k+c$, where $1\le c\le k-1$. For $1\le c\le k-2$, we obtain a tight spectral radius condition for a $(k+c)$-edge-connected graph to contain $k$ edge-disjoint spanning trees. We also obtain a tight spectral radius condition for $(2k-1)$-edge-connected graphs. In both cases, we give graph families containing all extremal graphs, and the graphs with maximum spectral radius in these families serve as the corresponding extremal graphs. Each graph in these families consists of a large clique and a small remaining part, with certain restrictions on the edges inside the small part and between the two parts. Moreover, for the case $m=k+1$, we further determine the unique extremal graph.

math.CO↗

arXiv2026

Size and spectral conditions for a graph with given minimum degree to be $k$-$d$-critical

A $k$-matching in a graph $G$ is defined as a function $f:E(G) \rightarrow \{0,1,\ldots,k\}$ satisfying $\sum_{e\in E_G(v)} f(e)$ $\leq k$ for each vertex $v\in V(G)$, where $E_G(v)$ denotes the set of edges incident to $v$ in $G$. For $1\leq d\leq k$ and $d \equiv |V(G)|~(\mathrm{mod}~2)$, if for any $ v \in V(G)$, there exists a $k$-matching $f$ such that $\sum_{e\in E_G(v)}f(e)=k-d$ and $\sum_{e\in E_G(u)}f(e)=k \text{ for any } u\in V(G)-\{v\}$, then $G$ is $k$-$d$-critical. A graph $G$ of odd order (resp. even order) is generalized factor-critical (resp. generalized bicritical) if the empty set is the unique set attaining the maximum value in $k$-Berge-Tutte-formula of $G$. In this paper, we provide sharp sufficient conditions in terms of size or spectral radius respectively for a graph $G$ to be $k$-$d$-critical, generalized factor-critical and generalized bicritical with minimum degree.

math.CO↗

arXiv2026

The Equivalence of Causal and Noncausal State Information on Bipartite Networks With State-Cognizant Receivers

State-dependent bipartite networks with state-cognizant receivers and state-informed transmitters are studied. Such networks have no nodes that both transmit and receive. Examples are the multi-access channel, the broadcast channel, and the interference channel. Without computing the capacity region of the network, it is shown that if the state sequence is ergodic and autonomous, and if, conditionally on the state sequence, the network law is memoryless, then the network capacity region does not depend on whether the state information is provided to the encoders causally or noncausally.

cs.IT↗

arXiv2026

Distance spectral radius conditions for perfect $k$-matching, generalized factor-criticality (bicriticality) and $k$-$d$-criticality of graphs

Let $G$ be a simple connected graph with vertex set $V(G)$ and edge set $E(G)$. A $k$-matching of a graph $G$ is a function $f:E(G)\rightarrow \{0,1,\ldots, k\}$ satisfying $\sum_{e \in E_G(v)} f(e) \leq k$ for every vertex $v \in V(G)$, where $E_G(v)$ is the set of edges incident with $v$ in $G$. A $k$-matching of a graph $G$ is perfect if $ \sum_{e \in E_G(v) } f(e) = k $ for any vertex $v \in V(G)$. The $k$-Berge-Tutte-formula of a graph $G$ is defined as: \[ \defk(G) = \max_{S \subseteq V(G)} \begin{cases} k \cdot i(G - S) - k|S|, & k \text{ is even;} \\[6pt] \odd(G - S) + k \cdot i(G - S) - k|S|, & k \text{ is odd.} \end{cases} \] A $k$-barrier of the graph $G$ is the subset $S \subseteq V(G)$ that reaches the maximum value in $k$-Berge-Tutte-formula. A connected graph $ G $ of odd (even) order is a {generalized factor-critical (generalized bicritical) graph about integer $ k $-matching}, abbreviated as a $ \mathrm{GFC}_k (\mathrm{GBC}_k)$ graph, if $\emptyset$ is a unique $k$-barrier. When $k$ is odd, let $ 1 \leq d \leq k $ and $ |V(G)| \equiv d \pmod{2} $. If for any $ v \in V(G) $, there exists a $ k $-matching $ h $ such that $\sum_{e \in E_G(v)} h(e) = k - d$ {and} $\sum_{e \in E_G(u)} h(e) = k$ for any $ u \in V(G) - \{v\} $, then $ G $ is said to be $ k $-$ d $-critical. In this paper, we provide sufficient conditions in terms of distance spectral radius to ensure that a graph has a perfect $k$-matching and a graph is $ k $-$ d $-critical, $\mathrm{GFC}_k$ or $\mathrm{GBC}_k$, respectively.

math.CO↗

arXiv2026

Size conditions and spectral conditions for generalized factor-critical (bicritical) graphs and $k$-$d$-critical graphs

Let $\mbox{odd}(G)$ and $i(G)$ denote the number of nontrivial odd components and the number of isolated vertices of a graph $G$, respectively. The $k$-Berge-Tutte-formula of a graph $G$ is defined as: $\mbox{def}_k(G)=\mathop{\text{max}}\limits_{S\subseteq V(G)}\{k\cdot i(G-S)-k|S|\} $ for even $k$; $\mbox{def}_k(G)=\mathop{\mbox{max}}\limits_{S\subseteq V(G)}\{\mbox{odd}(G-S)+k\cdot i(G-S)-k|S|\} $ for odd $k$. A $k$-barrier of a graph $G$ is the subset $S\subseteq V(G)$ that reaches the maximum value in the $k$-Berge-Tutte-formula of $G$. A graph $G$ of odd order (resp. even order) is generalized factor-critical (resp. generalized bicritical) if $\emptyset$ is its only $k$-barrier. Denote by $E_G(v)$ the set of all edges incident to a vertex $v$ in $G$. A $k$-matching of a graph $G$ is a function $f:E(G) \rightarrow \{0,1,...,k\}$ such that $\sum_{e\in E_G(v)} f(e)$ $\leq k$ for every vertex $v\in V(G)$. For $1\leq d\leq k$ and $d \equiv |V(G)|$(mod 2), if for any $ v \in V(G)$, there exists a $k$-matching $f$ such that $\sum_{e\in E_G(v)}f(e)=k-d$ and $\sum_{e\in E_G(u)}f(e)=k \text{ for any } u\in V(G)-\{v\}$. Then $G$ is $k$-$d$-critical. In this paper, we establish tight sufficient conditions in terms of size or spectral radius respectively for a graph $G$ to be generalized factor-critical, generalized bicritical, and $k$-$d$-critical. Furthermore, we prove the equivalence of the existence of four factors (namely, $\{K_2,\{C_t: t\geq 3\}\}$-factor, $\{K_2,\{C_{2t+1}:t\geq 1 \}\}$-factor, fractional perfect matching, perfect $k$-matching with even $k$) in a graph. Thus we also give size conditions and spectral radius conditions for a graph $G-v$ to have one of the four factors for any $v\in V(G)$.

math.CO↗

arXiv2026

Laplacian eigenvalue conditions for edge-disjoint spanning trees and a forest with constraints

Let $k$ be a positive integer and let $G$ be a simple graph of order $n$ with minimum degree $δ$. A graph $G$ is said to have property $P(k, d)$ if it contains $k$ edge-disjoint spanning trees and an additional forest $F$ with edge number $|E(F)| > \frac{d-1}{d}(|V(G)| - 1)$, such that if $F$ is not a spanning tree, then $F$ has a component with at least $d$ edges. Let $D(G)$ be the degree diagonal matrix of $G$. We denote $λ_i$ and $μ_i$ as the $i$th largest eigenvalue of the adjacency matrix $A(G)$ of $G$ and the Laplacian matrix $L(G) = D(G) - A(G)$ of $G$ for $i = 1, 2, \ldots, n$, respectively. In this paper, we investigate the relationship between Laplacian eigenvalues and property $P(k, δ)$. Let $t$ be a positive integer, and define $\mathcal{G}_t$ as the set of simple graphs such that each $G \in \mathcal{G}_t$ contains at least $t+1$ non-empty disjoint proper subsets $V_1, V_2, \ldots, V_{t+1}$ satisfying $V(G) \setminus \bigcup_{i=1}^{t+1} V_i \neq \emptyset$ and edge connectivity $κ'(G) = e(V_i, V(G) \setminus V_i)$ for any $i = 1, 2, \ldots, t+1$. For the class of graphs $\mathcal{G}_1$ with minimum degree $δ$, we provide a sufficient condition involving the third smallest Laplacian eigenvalue $μ_{n-2}(G)$ for a graph $G\in \mathcal{G}_1$ to have property $P(k, δ)$. Similarly, for the class of graphs $\mathcal{G}_2$ with minimum degree $δ$, we establish a corresponding sufficient condition involving the fourth smallest Laplacian eigenvalue $μ_{n-3}(G)$ for a graph $G\in \mathcal{G}_2$ to have property $P(k, δ)$. Furthermore, we extend the spectral conditions for all the results about $μ_{n-2}(G)$, $μ_{n-3}(G)$ and $λ_2(G)$ to the general graph matrices $aD(G) + A(G)$ and $aD(G) + bA(G)$.

math.CO↗

arXiv2026

Distance spectral radius conditions for edge-disjoint spanning trees and a forest with constraints

Let $k\ge 2$ be a positive integer and let $G$ be a simple graph of order $n$ with minimum degree $δ$. A graph $G$ is said to have property $P(k, d)$ if it contains $k$ edge-disjoint spanning trees and an additional forest $F$ with edge number $|E(F)| > \frac{d-1}{d}(n-1)$, such that if $F$ is not a spanning tree, then $F$ has a component with at least $d$ edges. Let $D(G)$ be the distance matrix of $G$. We denote $ρ_D(G)$ as the largest eigenvalue of $D(G)$, which is called the distance spectral radius of $G$. In this paper, we investigate the relationship between the distance spectral radius and the property $P(k, δ)$. We prove that for a connected graph $G$ of order $n \ge 2k+8$ with minimum degree $δ\ge k+2$, if $ρ_D(G) \le ρ_D(K_{k-1} \vee (K_{n-k} \cup K_1))$, then $G$ possesses property $P(k, δ)$. Furthermore, for a connected balanced bipartite graph $G$ of order $n \ge 4k+8$ with minimum degree $δ\ge k+2$, we show that if $ρ_D(G) \le ρ_D(K_{\frac{n}{2}, \frac{n}{2}} \setminus E(K_{1, \frac{n}{2}-k+1}))$, then $G$ also possesses property $P(k, δ)$. Our results generalize the work of Fan et al. [Discrete Appl. Math. 376 (2025), 31--40] from the existence of $k$ edge-disjoint spanning trees to the more refined structural property $P(k, δ)$.

math.CO↗

arXiv2025

Spectral extremal problems for $(a,b,k)$-critical and fractional $(a,b,k)$-critical graphs

A factor of a graph is essentially a specific type spanning subgraph. The study of characterizing the existence of $[a, b]$-factors based on eigenvalue conditions can be traced back to the work of Brouwer and Haemers (2005) on perfect matchings. With the advancement of graphs factor theory, the related spectral extremal problems, particularly the study of $[a,b]$-factors and fractional $[a,b]$-factors, have been widely studied by scholars. Our work is motivated by research related to the $[a,b]$-factors and fractional $[a,b]$-factors, and explores their generalizations: $(a,b,k)$-critical graphs and fractional $(a,b,k)$-critical graphs. A graph $G$ is called an $(a,b,k)$-critical (a fractional $(a,b,k)$-critical) graph if after deleting any $k$ vertices of $G$ the remaining graph of $G$ has an $[a,b]$-factor (a fractional $[a,b]$-factor). In this paper, we establish spectral radius conditions for a graph to be $(a,b,k)$-critical or fractional $(a,b,k)$-critical. When $k=0$, our results also resolve some open problems concerning $[a, b]$-factors and fractional $[a, b]$-factors.

math.CO↗

arXiv2025

Spectral radius and size conditions for fractional $(a,b,m)$-covered graphs

A fractional $(a,b,m)$-covered graph is a generalization of the concept of a fractional $[a,b]$-covered graph. For any $H \subseteq G$ with edge set $|E(H)| = m$, if there exists a fractional $[a,b]$-factor (the corresponding fractional indicator function is $h$) such that $h(e) = 1$ for any $e \in H$, then the graph $G$ is called a fractional $(a,b,m)$-covered graph. In this paper, we characterize the conditions for a graph to be a fractional $(a,b,m)$-covered graph from the perspectives of spectral radius and size, respectively.

math.CO↗

arXiv2025

An Efficient Secret Communication Scheme for the Bosonic Wiretap Channel

We propose a new secret communication scheme over the bosonic wiretap channel. It uses readily available hardware such as lasers and direct photodetectors. The scheme is based on randomness extractors, pulse-position modulation, and Reed-Solomon codes and is therefore computationally efficient. It is secure against an eavesdropper performing coherent joint measurements on the quantum states it observes. In the low-photon-flow limit, the scheme is asymptotically optimal and achieves the same dominant term as the secrecy capacity of the same channel.

quant-ph↗

arXiv2025

On the distance spectral radius, fractional matching and factors of graphs with given minimum degree

A fractional matching of $G$ is a function $f: E(G)\to [0,1]$ such that $\sum_{e\in E_G(v_i)}f(e)\le 1$ for any $v_i\in V(G)$, where $E_G(v_i)=\{e: e\in E(G) \ \textrm{and}\ e \ \textrm{is incident with} \ v_i\}$. Let $α_f(G)$ denote the fractional matching number of $G$, which is defined as $α_f(G)=\max\{\sum_{e\in E(G)}f(e): f\ \textrm{is a fractional matching of} \ G\}$. Let $\{G_1,G_2,G_3,\dots\}$ be a set of graphs, a $\{G_1,G_2,G_3,\dots\}$-factor of a graph $G$ is a spanning subgraph of $G$ such that each component of which is isomorphic to one of $\{G_1,G_2,G_3,\dots\}$. In this paper, we first establish a sharp upper bound for the distance spectral radius to guarantee that $α_f(G)>\frac{n-k}{2}$ in a graph $G$ of order $n$ with given minimum degree, where $0<k<n$ is an integer. Then we give a sharp upper bound on the distance spectral radius of a graph $G$ with given minimum degree $δ$ to ensure that $G$ has a $\{K_2, \{C_k\}\}$-factor, where $3\le k<+\infty$ is an integer. Moreover, we obtain a sharp upper bound on the distance spectral radius for the existence of a $\{K_{1,1},K_{1,2},\dots,K_{1,k}\}$-factor with $2\le k<+\infty$ in a graph $G$ with given minimum degree.

math.CO↗

arXiv2025

Matching, odd $[1,b]$-factor and distance spectral radius of graphs with given some parameters

For a connected graph $G$, let $μ(G)$ denote the distance spectral radius of $G$. A matching in a graph $G$ is a set of disjoint edges of $G$. The maximum size of a matching in $G$ is called the matching number of $G$, denoted by $α(G)$. An odd $[1, b]$-factor of a graph $G$ is a spanning subgraph $G_0$ such that the degree $d_{G_0}(v)$ of $v$ in $G_0$ is odd and $1\le d_{G_0}(v)\le b$ for every vertex $v\in V (G)$. In this paper, we give a sharp upper bound in terms of the distance spectral radius to guarantee $α(G)>\frac{n-k}{2}$ in an $n$-vertex $t$-connected graph $G$, where $2\le k \le n-2$ is an integer. We also present a sharp upper bound in terms of distance spectral radius for the existence of an odd $[1,b]$-factor in a graph with given minimum degree $δ$.

math.CO↗

arXiv2025

On the $D_α$ spectral radius of non-transmission regular graphs

Let $G$ be a connected graph with order $n$ and size $m$. Let $D(G)$ and $Tr(G)$ be the distance matrix and diagonal matrix with vertex transmissions of $G$, respectively. For any real $α\in[0,1]$, the generalized distance matrix $D_α(G)$ of $G$ is defined as $$D_α(G)=αTr(G)+(1-α)D(G).$$ The largest eigenvalue of $D_α(G)$ is called the $D_α$ spectral radius or generalized distance spectral radius of $G$, denoted by $μ_α(G)$. In this paper, we establish a lower bound on the difference between the maximum vertex transmission and the $D_α$ spectral radius of non-transmission regular graphs, and we also characterize the extremal graphs attaining the bound.

math.CO↗

arXiv2025

Tricyclic graphs for which the second largest distance eigenvalue less than $-\frac{1}{2}$

Let $G$ be a simple connected graph with vertex set $V(G)=\{v_{1}, v_{2}, \ldots, v_{n}\}$. The distance $d_G(v_i,v_j)$ between two vertices $v_i$ and $v_j$ of $G$ is the length of a shortest path between $v_i$ and $v_j$. The distance matrix of $G$ is defined as $D(G)=(d_G(v_i,v_j))_{n\times n}$. The second largest distance eigenvalue of $ G $ is the second largest eigenvalues of $D(G)$. Guo and Zhou [Discrete Math. 347(2024), 114082] proved that any connected graph with the second largest distance eigenvalue less than $-\frac{1}{2}$ is chordal, and characterize all bicyclic graphs and split graphs with the second largest distance eigenvalue less than $-\frac{1}{2}$. Based on this, we characterize all tricyclic graphs with the second largest distance eigenvalue less than $-\frac{1}{2}$.

math.CO↗