The graph coloring problem (GCP) is a classic combinatorial optimization problem that aims to find the minimum number of colors assigned to vertices of a graph such that no two adjacent vertices receive the same color. GCP has been extensively studied by researchers from various fields, including mathematics, computer science, and biological science. Due to the NP-hard nature, many heuristic algorithms have been proposed to solve GCP. However, existing GCP algorithms focus on either small hard graphs or large-scale sparse graphs (with up to 10^7 vertices). This paper presents an efficient hybrid heuristic algorithm for GCP, named HyColor, which excels in handling large-scale sparse graphs while achieving impressive results on small dense graphs. The efficiency of HyColor comes from the following three aspects: a local decision strategy to improve the lower bound on the chromatic number; a graph-reduction strategy to reduce the working graph; and a k-core and mixed degree-based greedy heuristic for efficiently coloring graphs. HyColor is evaluated against three state-of-the-art GCP algorithms across four benchmarks, comprising three large-scale sparse graph benchmarks and one small dense graph benchmark, totaling 209 instances. The results demonstrate that HyColor consistently outperforms existing heuristic algorithms in both solution accuracy and computational efficiency for the majority of instances. Notably, HyColor achieved the best solutions in 194 instances (over 93%), with 34 of these solutions significantly surpassing those of other algorithms. Furthermore, HyColor successfully determined the chromatic number and achieved optimal coloring in 128 instances.

Objective Many combinatorial optimization problems can be regarded as graph coloring problems. A classic topic in this field is total coloring, which combines vertex coloring and edge coloring. Previous studies and current research focus on the Total Coloring Conjecture (TCC), proposed in the 1960s. For graphs, including planar graphs, with maximum degree less than six, the correctness of the TCC has been verified through case enumeration. For planar graphs with maximum degree greater than six, the discharging technique has been used to confirm the conjecture by identifying reducible configurations and establishing detailed discharging rules. This method becomes limited when applied to planar graphs with maximum degree exactly six. Only certain restricted classes of graphs have been shown to satisfy the TCC, such as graphs without 4-cycles and graphs without adjacent triangles. More recent work demonstrates that the TCC holds for planar graphs without 4-fan subgraphs and for planar graphs with maximum average degree less than twenty-three fifths. Thus, it remains unclear whether planar graphs with maximum degree six that contain a 4-fan subgraph or have maximum average degree at least twenty-three fifths satisfy the conjecture. To address this question, this paper studies total coloring of a class of planar graphs known as nested n-pointed stars and aims to show that the TCC holds for these graphs. Methods The study relies on theoretical methods, including mathematical induction, constructive techniques, and case enumeration. An n-pointed star is obtained by connecting each edge of an n-polygon (n >= 3) to a triangle and then joining the triangle vertices not on the polygon to form a new n-polygon. Repeating this operation produces a nested n-pointed star with l layers, denoted by G(n)(l). These graphs have maximum degree exactly six. Their structural properties, including the presence of 4-fan subgraphs and maximum average degree greater than twenty-three fifths, are established. Induction on the number of layers is then used to show that G(n)(l) has a total 8-coloring: (1) G(n)(l) has a total 8-coloring; (2)G(n)(l -1) Suppose that has a total 8-coloring; (3) prove that has a G(l) (n) total 8-coloring. A graph G(n)(l) is defined as a type I graph if it has a total 7-coloring. When n= 3k , constructive arguments show that G(3k)(l) is a type I graph. The value of is considered in two cases, (k= 2m - 1)and (k = 2m) . In both cases, a total 7-coloring of G(3k)(l) is obtained by directly assigning colors to all vertices and edges. Results and Discussions Induction on the number of layers of G(n)(l) that nested n-pointed stars satisfy the Total Coloring Conjecture (Fig. 5). Five colors are assigned to the vertices and edges of to obtain a total 5-coloring (Fig. 6(a) and Fig. 8(a)). Two additional colors are then applied alternately to the edges connecting the polygons in layers 1 and 2. This produces a total 7-coloring of G(3k)(2) (Fig. 7(a) and Fig. 9(a)). After a permutation of the colors, another total 7-coloring of G(3k)(3) is obtained (Fig. 7(b) and Fig. 9(b)). The coloring pattern on the outermost layer is identical to that of G(3k)(1), which allows the same extension to construct total 7-colorings for G(3k)(4), G(3k)(5),... , G(3k)(l). Therefore, G(3k)(l) is a type I graph. Conclusions This study verifies that the Total Coloring Conjecture holds for nested n-pointed stars, which have maximum degree six and contain 4-fan subgraphs. It shows that G(3k)(l) is a type I graph. A further question arises regarding whether G(n)(l) is a type I graph when n not equal 3k. A total 7-coloring can be constructed when or n = 5, and therefore both G(4)(l) and G (l)(5) are type I graphs. For other values of n not equal 3k , whether G(n)(l) is a type I graph remains open.

We investigate the chromatic number problem, a classic NP-complete problem identified by Karp among his 21 seminal problems. The chromatic number of a graph G is the smallest integer k such that each vertex of G can be assigned one of k colors, with no two adjacent vertices assigned the same color. The chromatic number problem requires determining this minimum k for a given graph G with n vertices. The questions of whether an algorithm for the chromatic number problem with time complexity O*(a(n)), where a < 2, exists, and whether an algorithm for the chromatic number problem with time complexity O*(2(n)) and polynomial space exists, both remain unresolved. The fastest known algorithm for the chromatic number problem was proposed by Bj & ouml;rklund, Husfeldt, and Koivisto (FOCS 2006), with the time and space complexity of O*(2(n)). Subsequently, in their follow-up work (ICALP 2010), the space complexity is reduced to O(1.2916(n)). In this work, we present an improved algorithm for the chromatic number problem. Building on prior research, our approach leverages algebraic methods, specifically the generating functions and the discrete Fourier transform. Our main contribution demonstrates that by utilizing these algebraic techniques, certain structural properties of graphs can be exploited to reduce space complexity, while preserving the best-known time complexity of O*(2(n)). Specifically, our algorithm achieves a time complexity of O*(2(n)) and a space complexity of O*(2(9n/25) ) = O(1.2835(n)).

Discovering cohesive groups is a fundamental primitive in graph-based recommender systems, underpinning tasks such as social recommendation, bundle discovery, and community-aware modeling. In interaction graphs, cohesion is often modeled as the γ-quasi-clique, an induced subgraph whose internal edge density meets a user-defined threshold γ. This formulation provides explicit control over within-group connectivity while accommodating the sparsity inherent in real-world data. This paper presents EDQC, an effective framework for cohesive group discovery under explicit density constraints. EDQC leverages a lightweight energy diffusion process to rank vertices for localizing promising candidate regions. Guided by this ranking, the framework extracts and refines a candidate subgraph to ensure the output strictly satisfies the target density requirement. Extensive experiments on 75 real-world graphs across varying density thresholds demonstrate that EDQC identifies the largest mean γ-quasi-cliques in the vast majority of cases, achieving lower variance than the state-of-the-art methods while maintaining competitive runtime. Furthermore, statistical analysis confirms that EDQC significantly outperforms the baselines, underscoring its robustness and practical utility for cohesive group discovery in graph-based recommender systems.

Leaf chlorophyll (Chl) and carotenoids (Cxc) jointly govern light harvesting and photoprotection. However, concurrent retrieval from leaf reflectance remains challenging because Cxc absorption is weaker and strongly overlaps with Chl in the visible range. This spectral challenge, together with the physiological coordination between Chl and Cxc, motivates a multi-task learning (MTL) formulation for joint retrieval. Here, we propose a PROSPECT-guided MTL framework for the joint retrieval of leaf Chl and Cxc from 400–911 nm reflectance. The model is trained on PROSPECT-5 simulations and adopts a shared backbone with two pigment-specific branches. It is evaluated on five in-situ datasets (ANGERS, NX, BM, XS, and JTL; n = 680), benchmarked against separate single-task networks (STL), partial least squares regression (PLSR) and PROSPECT-based lookup-table inversion (PHY). Across datasets, the proposed MTL model consistently outperforms STL, PLSR and PHY for Chl and improves Cxc retrieval in most cases. In pooled evaluation, the RMSE for Chl is reduced by 32%, 13% and 28% compared with STL, PLSR and PHY, respectively. For Cxc, the corresponding reductions are 4%, 16% and 14%. On the public ANGERS benchmark, the model achieves an RMSE of 5.56 μg/cm2 for Chl (R2 = 0.93) and 2.39 μg/cm2 for Cxc (R2 = 0.78), yielding performance competitive with prior reflectance-based reports. PROSPECT-referenced occlusion analysis further suggests that, relative to STL, MTL induces more complementary wavelength reliance between the two tasks within pigment-sensitive regions. Overall, the PROSPECT–MTL framework enables accurate and robust reflectance-only joint retrieval of Chl and Cxc across datasets. These results provide a leaf-level basis for future canopy-scale and sensor-scale extensions in pigment-informed vegetation monitoring.

As transistor dimensions continue to shrink, binary devices are rapidly approaching their fundamental limits in power density. In response, multi-valued systems have attracted significant attention due to their enhanced information density. Among these, the ternary system stands out as the most practical option, being the closest integer base to (e), which is considered optimal for information efficiency. Despite the intrinsic advantages of DNA nanomaterials, such as programmability, energy efficiency, and massive parallelism, their application in ternary logic remains largely unexplored, particularly in the realm of ternary addition circuits. This gap can be attributed to a fundamental challenge: ternary logic requires circuits capable of recognizing and processing a far larger set of input combinations than binary systems, a task that existing models and techniques often struggle to accomplish. In this work, we propose a novel architecture for a ternary full adder. Our design includes a competitive blocking (CB) circuit that enables the recognition and computation of all possible three-input ternary combinations. Coupled with a dynamic concentration adjustment (CA) strategy, this approach significantly enhances the number of trits that can be processed. Biochemical experiments demonstrate that the CB circuit successfully yields the correct output digits for a ternary full adder, achieving 17-trit ternary addition. To our knowledge, this work represents the first successful DNA-based ternary adder, establishing a new methodological foundation for DNA computing and highlighting its considerable potential for scalable digital information processing.

From this chapter, we will present a series of methods to construct maximal planar graphs. In this chapter, we first introduce a simple but useful operation, called diagonal flip (or edge-flipping), by a finite sequence of which any two maximal planar graphs with the same order can be transformed into each other.

ObjectivesMutations in KIDINS220 are known to cause hereditary spastic paraplegia (HSP) and SINO syndrome. However, the phenotypic and genotypic spectrum of KIDINS220-related disorders remains incompletely understood. Herein, we describe the clinical, electrophysiological, histopathological, and genetic features of a novel KIDINS220 sterile alpha motif (SAM) -like domain mutation identified in a Chinese family with HSP accompanied by severe peripheral neuropathy (PN).MethodsClinical data, electrophysiological characteristics, and sural nerve histopathology were analyzed in a 19-year-old Chinese male. Genetic testing was performed in his family by using whole-exome sequencing, mitochondrial genome testing, and Sanger validation. A comprehensive literature review was conducted to analyze the phenotypic and genetic data of previously reported cases with KIDINS220 variants up to July 2025.ResultsThe proband exhibited classical signs of autosomal dominant HSP accompanied by severe multifocal sensory-motor PN. The spinal cord MRI showed mild spinal cord thinning, while the brain MRI and nerve ultrasound examinations were normal. Electrophysiological study revealed absent sensory nerve responses and globally reduced motor conduction velocities. Sural nerve biopsy confirmed significantly reduced nerve fiber density, myelin defects, axonal degeneration, and mitochondrial abnormalities. A heterozygous KIDINS220 c.3668A > G (p. Glu1223Gly) mutation, located within the SAM domain, was identified in both the proband and his mother. A total of 42 cases from 11 cohorts were reviewed.ConclusionWe suggest that patients with KIDINS220 SAM domain mutation may present with HSP accompanied by severe, mixed axonal and demyelinating PN, expanding the existing spectrum of the clinical phenotypes and pathogenic variants of KIDINS220.

Increasing research suggests that microRNAs (miRNAs) serve an essential function as biomarkers in various diseases. The variations in miRNA expression can influence their corresponding mRNAs, which, in turn, regulate the expression of target genes. Recently, graph neural networks (GNNs) have been widely utilized to predict miRNA-disease associations. However, a single GNN model is insufficient for fully learning node representations. Furthermore, individual aggregation methods struggle to effectively extract diverse structural information and node weights. To address these challenges, we propose a method that incorporates Principal Neighborhood Aggregation (PNA) and Graph Attention Networks (GAT) for miRNA-disease association prediction. First, we integrated multiple datasets to construct a weighted heterogeneous graph that models miRNA-LncRNA-disease interactions. Subsequently, PNA extracted node representations using multiple aggregators simultaneously. Additionally, features derived from both PNA and GAT were fused using an attention mechanism. These combined representations were then fed into a fully connected neural network for prediction. Experimental results demonstrate that PNAGMDA achieves exceptional performance, with AUC values of 93.82% and 92.77% on HMDD v2.0 and v3.2, respectively. Case studies, along with supplementary findings, confirm PNAGMDA's reliability for miRNA-disease prediction.

Drug-drug interactions (DDIs) present significant challenges within clinical pharmacology, as they can impact therapeutic outcomes, especially given the growing prevalence of polypharmacy. Traditional methods for the clinical validation of DDIs typically exhibit inefficiency and high cost, underscoring the necessity for more advanced computational methodologies. Although deep learning-based methods have improved DDI prediction performance, current approaches often face challenges in extracting and integrating multidimensional molecular features and capturing molecular reaction patterns. To overcome these limitations, we propose a Multidimensional Joint and Cross-learning (MDJCL) model that effectively integrates 1D, 2D, and 3D molecular features of drugs. A cross-attention fusion module aggregates multidimensional features while minimizing information loss, and a molecular-pair reaction module pinpoints potential interaction sites. Experimental results on benchmark data sets demonstrate that MDJCL consistently outperforms state-of-the-art models. Ablation studies reveal that each module contributes distinctively to the overall enhancement of evaluation metrics. These results validate the effectiveness of multidimensional feature integration and cross learning mechanisms in enhancing DDI prediction, offering a reliable tool for clinical decision-making and precision medicine.

Due to their exceptional programmability, DNA molecules are widely employed in the design of molecular circuits for applications such as DNA computing, DNA storage and cancer diagnosis and treatment. The quality of DNA sequences directly determines the reliability of these molecular circuits. However, existing DNA encoding algorithms suffer from limitations such as reliance on Hamming distance and conflicts among multiple objectives, resulting in insufficient stability of the generated sequences. To address these issues, this paper proposes a thermodynamics-based multi-objective evolutionary optimisation algorithm (TEMOA). The core innovations of the proposed algorithm are as follows: First, a thermodynamics-based DNA encoding modelling strategy (TDEMS) is introduced, which simplifies the encoding process and significantly improves the sequence quality by incorporating thermodynamic stability constraints. Second, two diversity optimisation strategies-the diversity assessment strategy (DAS) and the front equalisation nondominated sorting (FENS) strategy-are designed to enhance the algorithm's global search capability. Finally, a flexible fitness function design is incorporated to accommodate diverse user requirements. Experimental results demonstrate that TEMOA is more effective than state-of-the-art methods on challenging multi-objective optimisation problems, whereas the DNA sequences generated by TEMOA exhibit greater reliability compared to those produced by traditional DNA encoding algorithms.

The previous chapter introduced the enumerative DNA computation model for solving NP-complete problems, but with the increase of problem size, the amount of DNA molecules in the generated initial solution space will inevitably show an “exponential explosion”.

Accurately identifying protein-protein interaction sites (PPIS) is a critical challenge. Existing graph neural network (GNN) methods for PPIS prediction often overlook higher-order structural patterns. We propose ComGAT-PPIS, a Community-Augmented Graph Attention Network that addresses this limitation. Our model constructs a hierarchical graph by first detecting residue communities and then applies a graph attention mechanism across this community level before fusing features back to the residue level. Experiments on standard benchmarks show ComGAT-PPIS consistently outperforms state-of-the-art models, highlighting the importance of incorporating meso-scale topology for enhancing GNN-based PPIS prediction. Our code is available at https://github.com/BiscuitZhang/ComGAT.

Efficiently solving NP-complete problems-such as protein structure prediction, cryptographic decryption, and vulnerability detection-remains a central challenge in computer science. Traditional electronic computers, constrained by the Turing machine's one-dimensional data processing and sequential operations, struggle to address these issues effectively. To overcome this bottleneck, computational models must adopt multidimensional data structures and parallel information processing mechanisms. Building on our team's proposed probe machine model (a non-Turing computational framework), this study develops a blocking probe technique that leverages DNA computing's inherent parallelism to identify all valid solutions for NP-complete problems in a single probe operation. Using the 27-vertex 3-coloring problem as a case study, we successfully retrieved all solutions through DNA molecular probe experiments. This breakthrough demonstrates the first implementation of a fully parallel computing system at the molecular level, offering a novel paradigm for tackling computational complexity. Our results indicate that the probe machine, with its parallel architecture and molecular implementation, transcends the limitations of classical models and holds promise for solving intricate real-world problems.

Objective: Drug repositioning plays a pivotal role in expediting the drug discovery pipeline. The rapid development of computational methods has opened new avenues for predicting drug-disease associations (DDAs). Despite advancements in existing methodologies, challenges such as insufficient exploration of diverse relationships in heterogeneous biological networks and inadequate quality of negative samples have persisted. Methods: In this study, we introduce DRMGNE, a novel drug repositioning framework that harnesses metapath-guided learning and adaptive negative enhancement for DDA prediction. DRMGNE initiates with an autoencoder to extract semantic features based on similarity matrices. Subsequently, a comprehensive set of metapaths is designed to generate subgraphs, and graph convolutional networks are utilized to extract enriched node representations reflecting topological structures. Furthermore, the adaptive negative enhancement strategy is employed to improve the quality of negative samples, ensuring balanced learning. Results: Experimental evaluations demonstrate that DRMGNE outperforms state-of-the-art algorithms across three benchmark datasets. Additionally, case studies and molecular docking validations further underscore its potential in facilitating drug discovery and accelerating drug repurposing efforts. Conclusion: DRMGNE is a novel framework for DDA prediction that leverages metapath-based guidance and adaptive negative enhancement. Experiments on benchmark datasets show superior performance over existing methods, underscoring its potential impact in drug discovery.

DNA nanotechnology has created a wide variety of nanostructures that provide a reliable platform for nanofabrication and DNA computing. However, constructing programmable finite arrays that allow for easy pre-functionalization remains challenge. We aim to create more standardized and controllable DNA origami components, which could be assembled into finite-scale and more diverse superstructures driven by instruction sets. In this work, we designed and implemented DNA origami building block pieces (DOBPs) with eight mutually independent programmable edges and formulated DNA instructions that tailored such components. This system enables DOBPs to be assembled into one or more specific 2D arrays according to the instruction sets. Theoretically, a two-unit system can generate up to 48 distinct DNA arrays. Importantly, experiments results demonstrated that DOBPs are capable of both deterministic and nondeterministic assemblies. Moreover, after examining the effects of different connection strategies and instruction implementations on the yield of the target structures, we assembled more complex 2D arrays, including limited self-assembly arrays such as 'square frames', 'windmills' and 'multiples of 3' long strips. We also demonstrated examples of Boolean logic gates 'AND' and 'XOR' computations based on these assembly arrays. The assembly system provides a model nano-structure for the research on controllable finite self-assembly and offers a more integrated approach for the storage and processing of molecular information.

From 1994 to 2004, research in DNA computing was in its nascent stage, encompassing various aspects such as computational models, encoding, experimentation, and detection. Notably, during this period, the computational models were primarily enumerative in nature. These pioneering research outcomes not only laid the foundation for the deeper exploration of DNA computing but also provided a solid groundwork for RNA computing and, more broadly, the entire field of biological computing. This chapter focuses on the enumerative DNA computing models, selecting a subset of representative achievements for detailed introduction and in-depth analysis.

DNA computing is an emerging computational model that has garnered significant attention due to its distinctive advantages at the molecular biological level. Since it was introduced by Adelman in 1994, this field has made remarkable progress in solving NP-complete problems, enhancing information security, encrypting images, controlling diseases, and advancing nanotechnology. A key challenge in DNA computing is the design of DNA coding, which aims to minimize nonspecific hybridization and enhance computational reliability. The DNA coding design is a classical combinatorial optimization problem focused on generating high-quality DNA sequences that meet specific constraints, including distance, thermodynamics, secondary structure, and sequence requirements. This paper comprehensively examines the advances in DNA coding design, highlighting mathematical models, counting theory, and commonly used DNA coding methods. These methods include the template method, multi-objective evolutionary methods, and implicit enumeration techniques.

Real-time tactile perception with biological fidelity is critical for biomedical applications such as neural prosthetics and robotic surgeries, where sub-millisecond latency and micron-scale spatial resolution are essential. Event-based tactile sensors, inspired by mechanoreceptors, offer ultra-low latency and high energy efficiency but pose challenges for learning robust spatiotemporal representations under data scarcity and task variability. Current Spiking Graph Neural Networks (SGNNs) suffer from rigid spatial modeling and high computational costs, limiting deployment on edge devices. We propose MA-SGNN (Multi-view Adaptive SGNN), a lightweight brain-inspired framework emulating the biological tactile pathway: sensory encoding, feature extraction, and perceptual integration. MA-SGNN introduces: (1) a bio-hybrid spike encoder using Leaky Integrate-and- Fire neurons to capture temporal dynamics and extract biologically plausible features; (2) a multi-view adaptive graph constructor modeling structural and semantic taxel correlations via dynamic graphs; and (3) a spatiotemporal aggregator for efficient graph feature fusion. Evaluated on Ev-Objects and Ev-Containers benchmarks, MA-SGNN achieves competitive accuracy while reducing inference time by 59x and 90x versus state-of-the-art models. With only 10% training data, it maintains robust performance, dropping just 13.19%-significantly outperforming baselines. These results establish that MA-SGNN offers a biologically plausible and efficient solution for practical tactile intelligence.