Shi-Cai Gong

Let G be a simple graph with degree sequence (d1,d2,...,dn). The first and second Zagreb indices are defined respectively as the quantities M1(G) = ∑︁n i=1 d2 i and M2(G) = ∑︁ vivj∈E(G) didj. The Nordhaus-Gaddum-type invariant associated with the second Zagreb index is defined as follows ˜︂ M2(G) = M2(G)+M2(G), where G is the complement of G, and let Gn,m be the family of all graphs with n vertices and m edges. Let P(G) be a parameter of G. A graph G ∈ Gn,m is said to be P-maximal in Gn,m if P(G) = max{P(H) : H ∈ Gn,m}. Using an algebraic-combinatorial method, we in this paper show that, for any fixed (n,m), the extremal graphs in Gn,m maximizing M1 and ˜︂ M2 coincide.

In recent years, there has been a surge of interest in extremal problems concerning the enumeration of independent sets or cliques in graphs with specific constraints. For instance, the Kahn-Zhao theorem establishes an upper bound on the number of independent sets in a $d$-regular graph. Building on this, Cutler and Radcliffe extended the result by identifying the graph that maximizes the number of cliques among graphs with bounded order and maximum degree. In this paper, we introduce an innovative approach for counting cliques in graphs with a bounded maximum degree. To demonstrate the effectiveness of the method, we provide a new proof for the above Cutler-Radcliffe theorem and the Kahn-Zhao theorem.

Hypergraphs are powerful tools for modeling complex systems because they naturally encode higher-order interactions. However, most existing hypergraph representation-learning methods still struggle to capture such high-order structures, particularly in heterogeneous hypergraphs, which results in suboptimal performance on structure-sensitive tasks such as node classification. This paper presents WCRW-MLP, a new framework that integrates a Weighted- and Clustering-Biased Random Walk (WCRW) with a multi-layer perceptron. WCRW extends second-order random walks by introducing node-pair co-occurrence weights and triadic-closure clustering bias, enabling the walk to favor structurally significant and locally cohesive regions of the hypergraph. The resulting walk sequences are processed with Skip-gram to obtain high-quality structural embeddings, which are then concatenated with node attributes and fed into an MLP for classification. Experiments on several real-world hypergraph benchmarks show that WCRW-MLP consistently surpasses state-of-the-art baselines, validating both the efficacy of the proposed biasing strategy and the overall framework. These results demonstrate that explicitly modeling co-occurrence strength and local clustering is crucial for effective hypergraph embedding.

Citation networks are fundamental for analyzing the mechanisms and patterns of knowledge creation and dissemination. While most studies focus on pairwise attachment between papers, they often overlook compound relational structures, such as co-citation. Combining two key empirical features, superlinear node inflow and the temporal decay of node influence, we propose the Triangular Evolutionary Model of Superlinear Growth and Aging (TEM-SGA). The fitting results demonstrate that the TEM-SGA reproduces key structural properties of real citation networks, including degree distributions, generalized degree distributions, and average clustering coefficients. Further structural analyses reveal that the impact of aging varies with structural scale and depends on the interplay between aging and growth, one manifestation of which is that, as growth accelerates, it increasingly offsets aging-related disruptions. This motivates a degenerate model, the Triangular Evolutionary Model of Superlinear Growth (TEM-SG), which excludes aging. A theoretical analysis shows that its degree and generalized degree distributions follow a power law. By modeling interactions among triadic closure, dynamic expansion, and aging, this study offers insights into citation network evolution and strengthens its theoretical foundation.