In the k-diameter component connectivity model a network is consider operational if there is a component with diameter at least k. Therefore, a network is in a failure state if every component has diameter less than k. In this paper we find the vertex variant of the k-diameter component connectivity parameter, which is the minimum number of vertex deletions in order to put a network into a failure state, for particular classes of graphs. We also show the mixed variant by allowing vertex and edge failures within the network. We show results for paths, cycles, complete, and complete bipartite graphs for both variants as well as perfect r-ary trees for the vertex variant.

Identifying therapeutic target genes and their corresponding targeted drugs is of significant importance for the treatment of non-small cell lung cancer (NSCLC). This study proposes a multi-view graph auto-encoder model (MVGAE), which, together with the network-informed adaptive positive-unlabeled (NIAPU) and synthetic lethality multi-view graph auto-encoder (SLMGAE) model, constitutes an integrated computational framework. The framework integrates multi-source biological network data, including protein–protein interaction networks, disease-gene association information, and gene-drug bipartite graphs, for data mining. Through systematic analysis and computational screening, we ultimately predicted seven potential driver genes associated with NSCLC using the NIAPU model. The SLMGAE model predicted nine genes with synthetic lethality (SL) interactions to these driver genes as candidate therapeutic targets. Based on these SL targets, the MVGAE model further predicted corresponding targeted drugs. Notably, among the prioritized targets, existing studies indicate that ATR and RAD51 exhibit conditional SL effects in the context of functional impairment. Furthermore, several of the predicted candidate drugs (such as PAZOPANIB) have been previously reported to play a positive role in NSCLC treatment. This study highlights MVGAE as a novel computational framework for drug repurposing and demonstrates how its integration with complementary models can effectively prioritize potential therapeutic targets and candidate drugs, providing a robust computational basis for precision treatment strategies.

An effective two-stage auto-weighted multi-view consensus clustering via bipartite graph

Spectral conditions for k-extendability and k-factors of bipartite graphs

A note on almost cospectrality of components of NEPS of bipartite graphs

Robust adaptive anchor points and bipartite graph learning for image clustering

An innovative graph polynomial numerical strategy for the time-fractional Kuramoto-Sivashinsky model based on the complete bipartite graph's independence polynomials

Graph-theoretic degree-based descriptors play a central role in chemoinformatics and QSPR/QSAR modelling, yet most classical indices either focus purely on vertex degrees or treat bond contributions in a purely multiplicative way. In this work we introduce and systematically study a new family of modified bond-based indices in which each edge [Formula: see text] is weighted by a local bond factor [Formula: see text] in the denominator, coupled with a vertex kernel in the numerator. This construction yields modified versions of the first and second Zagreb indices, the Forgotten and Yemen indices, several connectivity-type descriptors (product, sum, Nirmala, ABC, CAB, GA, harmonic, and misbalance prodeg), as well as Sombor- and Dharwad-type bond indices. We first present a unified edge-partition representation for any symmetric kernel, expressing each modified index as a finite sum over degree classes [Formula: see text]. This framework allows us to derive closed-form expressions for all sixteen modified bond-based indices on a broad collection of benchmark families: paths [Formula: see text], cycles [Formula: see text], complete graphs [Formula: see text], complete bipartite graphs [Formula: see text], stars [Formula: see text], friendship graphs [Formula: see text], wheels [Formula: see text], book graphs [Formula: see text], Dutch windmill graphs [Formula: see text], and hypercubes [Formula: see text]. The resulting tables reveal clear asymptotic growth patterns and highlight which structures are extremal for the modified descriptors. Moreover, we obtain sharp degree-extreme bounds for a representative subset of the indices in terms of the order [Formula: see text], size m, and the minimum and maximum degrees δ and Δ, with equality characterizing regular graphs. The proposed modified bond-based indices thus provide a flexible and analytically tractable family of descriptors that couple vertex and bond information in a novel way, and are well suited as structured features for modern chemoinformatics and graph-based machine-learning models on molecular graphs. Finally, to demonstrate predictive utility in a hypothesis-driven setting, we further benchmark these [Formula: see text] descriptors within a large multi-task QSAR/QSPR pipeline on 3,219 ChEMBL antibacterial molecules across ten continuous properties using a heterogeneous model zoo under three descriptor scenarios, where the combined descriptors scenario achieves the best overall generalisation (Macro Test [Formula: see text]; Global zRMSE [Formula: see text]), improving upon the Physicochemical descriptors scenario (Macro Test [Formula: see text]; Global zRMSE [Formula: see text]).

ABSTRACT Let be a finite, loopless graph that may contain multiedges. We call a ring graph if is obtained from a cycle by duplicating some edges. Denote by and the chromatic index and maximum degree of , respectively. Kőnig's classical result implies that if is a bipartite graph. Goldberg showed that , where is the length of a shortest odd cycle of . Stiebitz, Scheide, Toft, and Favrholdt conjectured that if reaches this upper bound, then contains a ring graph as a subgraph with the same chromatic index. Cao, Chen, He, and Jing found some counterexamples for the conjecture. In this paper, we establish a necessary and sufficient condition for the conjecture of Stiebitz et al. to hold. More specifically, writing with in the division‐remainder form, we show that if then the conjecture holds, otherwise the conjecture fails. If graph has an odd number of vertices, a matching of covering all, but one, vertices of is called a near‐perfect‐matching of . We characterize ring graphs, as well as ‐graphs, that have a near‐perfect‐matching factorization, and use these decomposition theorems to obtain the above characterization of the truth of the conjecture of Stiebitz, Scheide, Toft, and Favrholdt.

Chemical graph theory is an interdisciplinary field that integrates fuzzy-graph-theoretical and computational methods to model molecular structures as fuzzy graphs and to address associated mathematical problems. Topological indices assign numerical invariants to network structures; among them, the Sombor index, originally introduced in chemical graph theory, provides a useful measure for quantifying the structure of molecular graphs within a fuzzy-graph framework. In this paper, we investigate bounds for the Sombor index across several graph families and operations, including edge addition and deletion, broom graphs, fuzzy star graphs on n vertices, complete bipartite fuzzy graphs, and the star graph (K 1,t ). We also examine the relationship between the Sombor index and various properties of alkanes and octane isomers. Furthermore, we demonstrate a significant correlation between this index and multiple thermodynamic parameters, such as heat of vaporization, entropy, acentric factor, and enthalpy of vaporization, while observing a relatively weak correlation with the heat capacity of octane isomers. Finally, we apply the Sombor index in fuzzy graphs (SOF([Formula: see text] )) to identify and rank Indian states according to their crime rates. These results highlight the broad applicability of the Sombor index in both chemical graph theory and real-world network analysis.

We present a theorem which characterizes the class of line graphs of directed graphs. The characterization is an analogue of both the characterization of line graphs by Krausz (1943) and of directed line graphs of directed graphs by Harary and Norman (1960). Our characterization simplifies greatly in the case that the graph is bipartite. This and another result which we present draws attention to the special case of bipartite line graphs of directed graphs. As a result we explore the problem of finding the complete list of forbidden subgraphs for the class of bipartite line graphs of directed graphs. It appears, however, that this problem is extremely difficult. We do find two infinite families of forbidden subgraphs as well as several other illustrative examples.

Structure sparsity-induced bipartite graph learning for multi-view clustering

Graph Neural Networks (GNNs) have become foundational models in recommender systems due to their ability to propagate information over user–item bipartite graphs via neighborhood aggregation. Despite their empirical success, GNNs are inherently constrained by their reliance on local connectivity, which limits their ability to capture global interaction patterns, particularly in large-scale recommendation scenarios characterized by severe data sparsity. To address these challenges, we propose the Taylor Linear attention in Transformer (TLFormer), which enhances recommendation performance by enabling global attention across all user–item pairs while preserving graph structural information. Unlike existing Transformer-based recommendation approaches that focus on local attention patterns, TLFormer introduces a novel linear attention mechanism derived from the first-order Taylor approximation, allowing efficient computation of all-pair interactions. TLFormer integrates spatial topology as positional encoding while maintaining linear complexity, effectively balancing computational efficiency with model expressiveness for large-scale recommendation scenarios. Extensive experiments across multiple datasets demonstrate that TLFormer significantly outperforms state-of-the-art methods, particularly in scenarios with sparse interactions and long-tail distributions.

A bstract Brane tilings are bipartite periodic graphs on the 2-torus and realize a large family of 4 d 𝒩 = 1 supersymmetric gauge theories corresponding to toric Calabi-Yau 3-folds. We present a complete classification of dimer integrable systems corresponding to the 30 brane tilings whose toric Calabi-Yau 3-folds are given by the 16 reflexive polygons in 2 dimensions. For each dimer integrable system associated to a reflexive polygon, we present the Casimirs, the single Hamiltonian built from 1-loops, the spectral curve, and the Poisson commutation relations. We also identify all birational equivalences between dimer integrable systems in this classification by presenting the birational transformations that match the Casimirs and the Hamiltonians as well as the spectral curves and Poisson structures between equivalent dimer integrable systems. In total, we identify 16 pairs of birationally equivalent dimer integrable systems which combined with Seiberg duality between the corresponding brane tilings form 5 distinct equivalence classes. Echoing phenomena observed for brane brick models realizing a family of 2 d (0 , 2) supersymmetric gauge theories corresponding to toric Calabi-Yau 4-folds, we illustrate that deformations of brane tilings, including mass deformations, correspond to the birational transformations we discover in this work, and leave invariant the number of generators of the mesonic moduli space as well as the corresponding U(1) R -refined Hilbert series.

Multi-view spectral clustering algorithm based on bipartite graph and multi-feature similarity fusion.

Dense subgraph mining in dynamic bipartite graphs

Unsupervised multi-view feature selection via attentive hierarchical bipartite graphs with optimizable graph filter

Anchor point segmentation based multi-view clustering.

Let L be a finite lattice with two atoms. In this paper we have shown existence of sublattice N such that N and L have two atoms. Studied zero-divisor graph of a lattice L and N. Further, explored relation between both zero divisor graphs and they are complete bipartite graph.

Whether learners can correctly complete exercises is influenced by multiple factors, including their mastery of relevant knowledge concepts and the interdependencies among these concepts. To investigate how the structure of the knowledge space—particularly the complex relationships among learners, exercises, and knowledge points—affects learning outcomes, this study proposes the Hierarchical Heterogeneous Graph Knowledge Tracing model (HHGKT). A hierarchical heterogeneous graph was constructed to capture two types of interactions—“learner–knowledge concept” and “exercise–knowledge concept”—and incorporate the interdependencies among knowledge concepts into the graph structure. By leveraging this hierarchical representation, the model’s ability to characterize learners and exercises was enhanced. A hierarchical heterogeneous graph encompassing users, exercises, and knowledge concepts was built based on the ASSISTments dataset, and simulation experiments were conducted. The results indicate that the proposed structure effectively represents the complexity of the knowledge space. Incorporating knowledge concept interdependencies improves prediction accuracy by 1.79%, while the hierarchical heterogeneous graph outperforms traditional bipartite graphs by approximately 1.5 percentage points in accuracy. These findings demonstrate that the model better integrates node and relational information, offering valuable insights for knowledge space modeling and its application in educational contexts.