Current Issue

2025, Vol. 45 No. 3   Published date: 28 September 2025
Previous Issue   

• Large Prey Growth in the Wolkowicz-Rothe-Shafer Predator-Prey System with Group Defense

  André ZEGELING, LIAO Jin

  2025, 45(3): 1-52. doi: 10.3969/j.issn.1006-8074.2025.03.001

  Abstract ( 377 )   PDF (4842KB) ( 56 )   　　

  This paper investigates the number of limit cycles in a predator-prey system with group defense, intially introduced by Wolkowicz and later examined by Rothe and Shafer in the 1980's. Under the assumption of large prey growth, the system reduces to a perturbed singular system, whose limit cycles can be analyzed using geometric singular perturbation methods—primarily through

  the study of a slow-divergence integral. Our work completes partially the results previously obtained by Li and Zhu and by Hsu. We provide a comprehensive classification of all possible singular cycles capable of generating limit cycles and analyze the slow-divergence integral for the nine distinct types of cycle families that arise in a canard explosion. Based on these findings, we demonstrate that the maximum number of limit cycles emerging from the singular cycles is two in all cases, thereby confirming conjectures posed by Rothe-Shafer and Xiao-Ruan.

• Pfaffian Property and Planarity of Cayley Graphs on Dicyclic Groups

  TANG Lang, LIU Weijun, LU Rongrong

  2025, 45(3): 53-65. doi: 10.3969/j.issn.1006-8074.2025.03.002

  Abstract ( 431 )   PDF (6709KB) ( 32 )   　　

  The Pfaffian property of graphs is of fundamental importance in graph theory, as it precisely characterizes those graphs for which the number of perfect matchings can be computed in polynomial time with respect to the number of edges. The study of Pfaffian graphs originated from the enumeration of perfect matching in planar graphs. References \cite{4,5,7} demonstrated that every planar graph is Pfaffian. Therefore, the Pfaffian property and planarity of graphs play a vital role in modern matching theory.

  This paper contributes a complete characterization of the Pfaffian property and planarity of connected Cayley graphs over the dicyclic group $T_{4n}$ of order $4n$ $(n\geq 3)$, shows that the Cayley graph $Cay(T_{4n}, S)$ is Pfaffian if and only if $n$ is odd and $S=\{a^{k_1},a^{2n-k_1},ba^{k_2},ba^{n+k_2}\}$, where $1\leq k_1\leq n-1$, $0\leq k_2\leq n-1$ and $(k_1,n)=1$, and furthermore, shows that $Cay(T_{4n}, S)$ is never planar.

• On the Extremal Values of the Sombor Index for Halin Graphs

  LI Yunping , TANG Zikai

  2025, 45(3): 66-80. doi: 10.3969/j.issn.1006-8074.2025.03.003

  Abstract ( 384 )   PDF (538KB) ( 36 )   　　

  Let $G$ be a simple connected graph with vertex set $V(G)$ and edge set $E(G)$. Then the Sombor index of graph $G$ is defined as $SO(G)=\sum_{uv\in E(G)}\sqrt{d^2(u)+d^2(v)}$, where $d(u)$ denotes the degree of vertex $u$. In this paper, the maximum and minimum values of the Sombor index for Halin graphs are obtained, and the corresponding extremal graphs are characterized.

• Precise Value of the Invariant $s_{d\mathbb{N}} (D_{2n})$ over the Dihedral Group $D_{2n}$

  ZHAO Kevin, LIANG Wanjun, CHEN Lifang

  2025, 45(3): 81-95. doi: 10.3969/j.issn.1006-8074.2025.03.004

  Abstract ( 439 )   PDF (215KB) ( 35 )   　　

  Let $G$ be a finite group and $d$ a positive integer. Let $s_{d\mathbb{N}} (G)$ denote the smallest positive integer $l$ such that every sequence over $G$ of length at least $l$ contains a nonempty product-one subsequence $T$ with $|T|\equiv0 \pmod{d}$. This paper studies $s_{d\mathbb{N}} (D_{2n})$ for the dihedral group $D_{2n}$ and shows that when $n=2^r$ with $r\geq3$, the equality $s_{d\mathbb{N}} (D_{2n})=\operatorname{lcm}(n, d)+\gcd(n, d)$ holds.

• Normalized Solutions for a Logarithmic Schrödinger Equation

  LIU Xiang, LEI Chunyu

  2025, 45(3): 96-106. doi: 10.3969/j.issn.1006-8074.2025.03.005

  Abstract ( 409 )   PDF (166KB) ( 44 )   　　

  In this paper, we consider the following logarithmic Schrödinger equation:

  \begin{equation*}

  -\Delta u+\omega u =u\log|u|^2,~~ u\in H^1(\mathbb{R}^N),

  \end{equation*}

  where $N\geq3$, and $\omega>0$ is a constant. With an auxiliary equation, we obtain the existence of normalized solutions by using the constrained variational method.

• Study on Cascading Failures Based on Intra-Layer and Inter-Layer Structures of Multiplayer Networks

  CHEN Mengjiao, WANG Niu, WEI Daijun

  2025, 45(3): 107-124. doi: 10.3969/j.issn.1006-8074.2025.03.006

  Abstract ( 406 )   PDF (4338KB) ( 51 )   　　

  Compared to single-layer networks, multilayer networks exhibit a more complex node degree composition, comprising both intra-layer and inter-layer degrees. However, the distinct impacts of these degree types on cascading failures remain underexplored. Distinguishing their effects is crucial for a deeper understanding of network structure, information propagation, and behavior prediction. This paper proposes a capacity-load model to influence and compare the influence of different degree types on cascading failures in multilayer networks. By designing three node removal strategies based on total degree, intra-layer degree, and inter-layer degree, simulation experiments are conducted on four types of networks. Network robustness is evaluated using the maximum number of removable nodes before collapse. The relationships between network robustness and the coupling coefficient, as well as load and capacity adjustment parameters, are also analyzed. The results indicate that the node removal strategy with the least impact on cascading failures varies across different types of networks, revealing the significance of different node degrees in failure propagation. Compared to other models, the proposed model enables networks to maintain a higher maximum number of removable nodes during cascading failures, demonstrating superior robustness.