Search Results (81)

The Hyperbolic Sombor index $HSO\left(G\right)$ of a graph $G=\left(V\right(G),E(G\left)\right)$ is proposed as a new vertex-degree-based topological index. In this paper, we investigate the mathematical properties of this novel vertex-degree-based topological index for general graphs. We first establish tight upper and lower bounds on $HSO\left(G\right)$ in terms of the variable Euler–Sombor index $EU(\lambda ,G)$ for all $\lambda \ge -2$. Furthermore, we derive new bounds connecting $HSO\left(G\right)$ with three other well-known degree-based invariants: the second Zagreb index $M_2\left(G\right)$, the Elliptic Sombor index $ESO\left(G\right)$, and the Forgotten Sombor index $FSO\left(G\right)$. Full article

Accurately identifying critical nodes in complex networks and applying targeted protection strategies significantly enhances network security. Traditional importance metrics rely on a single topological feature and cannot fully capture node influence. Existing multi-attribute fusion methods integrate multiple structural sources but typically use fixed weights or predefined rules, failing to adaptively adjust attribute contributions based on local and global network characteristics, which limits their generalization across diverse networks. To address this, we propose the CRITIC-based Objective Weighting and Multi-Metric Fusion Method (COWMF). COWMF first builds a Graph Attention Network with Virtual Global–Local Integration (GAT-VGL), taking four low-complexity topological metrics, degree centrality (DC), H-index, degree and neighborhood information centrality (DNC), and k-shell, as input. Through a learnable attention mechanism, GAT-VGL adaptively aggregates multi-hop neighborhood information and explicitly incorporates global structural information via a virtual node to achieve whole-graph topological awareness, generating a global influence score with good discriminative power and high computational efficiency. This score is then integrated with DC and DNC into an improved CRITIC-based objective weighting fusion scheme, enabling adaptive synergy among local connectivity, semi-local radiation, and global structure. Experiments on six real-world networks of varying types and scales show that COWMF demonstrates relatively stable and competitive performance in both simulated attack and susceptible-infected-recovered (SIR) spreading simulations, two complementary experiments, demonstrating satisfactory disruptive capability and propagation influence. Its importance scores exhibit high monotonicity across all networks, with good discriminative power. Full article

This paper proposes a resilience assessment model for irregular intersections that combines conflict analysis, channelization conditions, and traffic operational data to identify turning movements with high cascading-failure potential. Turning movements are the basic analytical unit. A channelization-weighted conflict matrix captures coupling among traffic flows by incorporating signal phase separation and lane function allocation. A CLI (Conflict Load Index) integrates the weighted conflict degree, a saturation correction factor, and expert risk scores to rank turning movements by cascading-failure risk. Three failure scenarios are compared: random failure, descending conflict-degree failure, and descending CLI failure. A failure propagation probability function based on weighted conflict-degree load distribution is paired with a resilience loss index that quantifies cumulative intersection capacity loss during failure propagation. The model is applied to the irregular intersection near Donggang Passenger Station. Among 16 turning movements, R2-T2, R4-T2, R5-T2, and R1-T1 have the four highest CLI values and form the high-risk set. The high-saturation movement R5-T2 (saturation 0.738) rises to rank 3 in the revised model, three positions higher than in the topology-only model, reflecting its risk level under actual traffic conditions. Sensitivity analysis shows that the relative ranking of the three failure modes is consistent across all parameter combinations, confirming the robustness of the conclusions. The model provides a quantitative basis for resilience diagnosis, risk early warning, and improvement planning at irregular intersections. Full article

Topological indices provide numerical descriptions of graph structure and support graph analysis in cheminformatics, network design, and graph mining. This study presents a reproducible computational framework that combines controlled cyclic-graph generation, structure-preserving transformations, exact computation of six classical topological indices, and multi-output graph neural network regression. The framework evaluates the Wiener, Merrifield–Simmons, Hosoya, first Zagreb, second Zagreb, and Randi’c indices for unicyclic and bicyclic graphs. It represents each graph using sparse connectivity and node-level features that encode degree, cycle membership, pendant connectivity, leaf status, and normalized eccentricity. A graph isomorphism network (GIN) jointly predicts the six indices and is compared with graph convolutional networks (GCNs), graph attention networks (GATs), and descriptor-based regression baselines. The controlled benchmark shows that nonlinear descriptor-based models achieve the lowest aggregate errors because the supplied graph-level descriptors contain strong prior information about graph size, degree structure, branching, and cycle complexity. Although GIN does not achieve the highest overall accuracy, it provides the strongest graph-native performance by learning directly from sparse connectivity and node-level features without requiring a fixed handcrafted graph-level descriptor vector. The proposed surrogate does not replace exact evaluation for isolated small graphs, where exact computation remains more appropriate. Instead, its practical value emerges through repeated evaluations of larger, more complex graph instances. To examine this setting, a computational stress experiment evaluates sparse multicyclic graphs under increasing cyclomatic complexity and measures exact computation time, timeout frequency, prediction accuracy, and the amortized break-even point. The results indicate that surrogate prediction becomes beneficial when combinatorial index computation becomes sufficiently expensive, and the trained model is reused across many structurally related graph queries. An external experiment on circulant graphs also demonstrates that the framework can extend beyond the original graph generators by modifying only the graph-construction stage. Full article

The Sombor index has recently become a central tool among degree-based graph invariants; however, it does not explicitly isolate degree imbalance along edges. In this work, we introduce the degree-based Contrastive Sombor Index (CSO), which combines endpoint-degree magnitude with local degree imbalance. For a finite simple graph $G=(V,E)$, the index is defined by $\mathrm{CSO}\left(G\right)={\displaystyle \sum _{uv\in E\left(G\right)}}\left(\sqrt{d{\left(u\right)}^2+d{\left(v\right)}^2}-\sqrt{2}\min \{d\left(u\right),d\left(v\right)\}\right).$ Unlike the Sombor index, which primarily reflects the magnitude of the endpoint degrees, the CSO contribution vanishes when the endpoint degrees are equal and responds to degree imbalance while retaining degree-scale information. In particular, it can distinguish certain graphs having the same total edgewise irregularity but different endpoint-degree distributions. In this work, we first show that $CSO\left(G\right)\ge 0$ and prove that $CSO\left(G\right)=0$ if and only if each connected component of G is regular. We also establish general lower and upper bounds for CSO. In addition, we obtain a relation connecting the CSO index with the first Zagreb index and the edgewise degree differences. We also discuss extremal aspects of the index. As an application, we derive an explicit summation formula for CSO on monogenic semigroup graphs. From our computations on $\Gamma \left(S_M\right)$, it follows that the asymptotic growth order of the index satisfies $CSO(\Gamma \left(S_M\right))=\Theta \left(n^3\right).$ These results show that the CSO index combines degree-magnitude information with sensitivity to unequal endpoint degrees and provides an additional perspective on degree heterogeneity in graphs. Full article

Control subarea division is essential for coordinated signal control, but methods based mainly on local correlation or static topology may not adequately capture traffic-state propagation under dynamic traffic loading. This study proposes a control subarea division method that explicitly models traffic-state propagation by integrating state-guided colored random walk and path entropy analysis. Intersection correlation degree and traffic state are used to construct a state-guided colored random walk process, in which transition probabilities are updated according to network connectivity and traffic-state consistency. Path entropy characterizes propagation uncertainty, and control subareas are identified by minimizing the distribution discrepancy between node-level and subarea-level path responses. To compare partitioning schemes, five complementary metrics were adopted: variance reduction rate of spatial delay, delay reduction rate, congestion mitigation index, stop reduction rate, and queue reduction rate. A VISSIM microsimulation model with dynamic traffic loading was developed to compare the proposed method with the Whitson and Fast Newman methods. The proposed method achieved the best performance across all five metrics, with values of 41.47%, 23.77%, 25.96%, 23.59%, and 15.08%, respectively. These results indicate that the proposed method improves spatial balance and network efficiency while mitigating bottlenecks, reducing stops, and suppressing queue accumulation. Full article

The diminished Sombor index is a degree-based topological index that normalizes the Sombor contribution of each edge (defined as the Euclidean norm of the endpoint degrees) by the sum of those degrees, thereby making the index independent of graph size and ensuring a more balanced reflection of the relative degree contributions of adjacent vertices. In this paper, we investigate the extremal behavior of the diminished Sombor index over the class of connected bipartite graphs with fixed order and diameter. We establish a sharp upper bound for this index within the family of all bipartite graphs on a given number of vertices and with a prescribed diameter, and we completely characterize the extremal graphs attaining this bound. Furthermore, we prove that the maximum diminished Sombor index strictly decreases as the diameter increases. As a consequence, we determine the connected bipartite graphs of fixed order that achieve the three largest values of the diminished Sombor index. Full article

Zagreb-type indices are topological indices derived from the degrees of nodes. The first Zagreb index, the F-index, and the Y-index represent the sum of the squares, cubes, and fourth powers of all node degrees, respectively. These indices are valuable for understanding the chemical reactions, physical characteristics, and biological activities of various substances. In this study, we explore the connection between Y-index and the graph Laplacian spectrum. Additionally, we introduce the fractal graphs based on star graphs, a class of extended Vicsek graphs, and derive the rules for eigenvalue evolution between two generations of the graph. Ultimately, we provide exact closed-form expressions for the first Zagreb index, F-index, and Y-index of the fractal graphs based on star graphs by using spectral graph theory. Full article

The global shipping network, which handles over 80% of international trade volume, is increasingly exposed to disruptions from typhoons and other extreme weather events under climate change. However, conventional network vulnerability assessments often overlook the geographically heterogeneous nature of such natural hazards. Here, we introduce a typhoon-related systemic vulnerability model (GMSN-TV) that integrates three core components: typhoon exposure, port network sensitivity, and national adaptive capacity, to quantify the Typhoon Vulnerability Index (TVI) of 1075 major ports across 2017 and 2021. Our analysis reveals four key findings. First, the global shipping network became structurally sparser between 2017 and 2021, with edges declining by 17.84% and network efficiency decreasing by 4.22%, rendering it more susceptible to climate-related disruptions. Second, simulated TVI-based natural attacks and conventional degree-based deliberate attacks induce fundamentally different risk patterns: removing the top 10% high-TVI ports in 2021 caused a 6.3% decline in network efficiency, whereas removing the top 10% hub ports resulted in a 20.1% decline, a difference of 13.8 percentage points; however, natural attacks proved more effective at isolating peripheral ports, generating an isolated node ratio of 1.16% compared to 0.00% under deliberate attacks. Third, when removing the top 50% high TVI ports, the contribution of typhoon vulnerability to network degradation increased from 13.77% in 2017 to 15.87% in 2021. Fourth, high-vulnerability ports exhibit significant spatial clustering, with the Northwest Pacific region (50.8%) and the North Atlantic region (29.5%) collectively accounting for over 80% of global high-vulnerability ports in 2021. Compared to conventional topology-based assessments, the GMSN-TV analytical framework proposed in this study integrates typhoon hazard data with network topology, providing a novel scientific tool with enhanced identification efficacy and accuracy. It successfully captures local network disintegration effects entirely missed by traditional deliberate attacks, revealing an isolated node ratio of 12.5% after removing 70% of high-TVI ports. This demonstrates the tool’s precision in identifying hidden high-risk peripheral nodes, enabling decision-makers to prioritize climate adaptation investments for critical maritime infrastructure more accurately. Full article

Topological graph theory provides a quantitative approach to understanding the structural complexities of sulfonamide compounds, which are prominent for their therapeutic importance in cancer treatment. A new computational scheme to predict the physicochemical and biological functions of sulfonamide derivatives, based on connection numbers and connection-based topological indices as alternatives to the theoretically overt degree-based index, is proposed. A set of structurally diverse sulfonamide compounds as chemical graphs is considered, and the relevant graph descriptors are computed using different connection numbers. Due to the complexity of the calculations involved in connectivity and other such indices, algorithms were developed in Python 3.12.12 to automate the extraction and calculation of these indices. QSPR analysis, with the help of supervised machine learning models like linear regression, among others, and various statistical techniques, was employed to obtain insight into the relationships existing between the structural properties and the molecular properties measured, such as melting point, molecular weight, etc. These results demonstrate the great predictive capability of connection-based indices in assessing pharmacologic efficacy or molecular behavior. The holistic setting thus links topological modeling to data-driven prediction and provides a window into the rational design and optimization of sulfonamide-based cancer therapeutics. Full article

A novel approach for analyzing the structural integrity and operational vulnerability of complex networks using intuitionistic fuzzy graphs has been modeled. While traditional fuzzy graph metrics focus primarily on existence, they fail to capture the holistic systemic impact of failures. To overcome this limitation, a scalar-based measure of nodal importance that integrates both existence (membership degree) and non-existence (non-membership degree) values of incident edges into a single critical metric has been developed. The proposed indices demonstrate enhanced sensitivity to network perturbations compared to conventional degree centrality measures, capturing latent vulnerabilities in critical infrastructure topologies. Based on this, two indices are proposed: Intuitionistic Fuzzy Degree Index and Intuitionistic Edge Interaction Index. These indices quantify the total system activity, stress dispersion, overall network cohesiveness, and potential for cascading failure propagation. When applied to synthetic core-periphery networks, the proposed indices identified critical nodes with superior discrimination capability compared to existing fuzzy graph metrics, revealing that removal of identified nodes results in system-wide connectivity degradation observable through both membership and non-membership approximations. This methodology was applied to a core-periphery communication network to analyze the systemic consequences of node removal. Experimental validation on networks of varying sizes demonstrates that the Intuitionistic Edge Interaction Index achieves robust node criticality ranking across heterogeneous network topologies with improved predictive accuracy for cascade initiation points. This work provides network analysts and engineers a quantitative tool to precisely assess criticality and inform targeted resilience strategies in uncertain, high-risk environments. Full article

The H-index is a widely recognized centrality measure for nodes in symmetric networks, defined as the maximum number of neighbors with degrees equal to or greater than the node’s own degree. However, this metric underestimates the structural influence of “weak nodes”—low-degree nodes connected to high-degree hubs—that often serve as critical connectors in network topology. To address this limitation, we propose the $H_{\alpha }$-index, which generalizes the H-index by considering the maximum number of neighbors with degrees at least $\alpha$ times the node’s degree, where $\alpha \ge 1$. Based on this refinement, we introduce two novel centrality measures: the g-core and the local g-core, which were derived from iterative applications of the $H_{\alpha }$-index to a node’s neighbors. Extensive experiments on sixteen real-world networks demonstrate the efficiency of our methods. Notably, the local g-core achieves 45–105% higher Kendall Tau correlation coefficients compared to the traditional H-index and coreness on three benchmark networks, highlighting its superior performance in capturing node influence. Full article

Aiming at the strong spatio-temporal coupling relationship between data in the actual industrial production process, which leads to the problem of insufficient reliability and poor timeliness of traditional process anomaly monitoring methods, a time series anomaly monitoring model based on the graph similarity network with multi-scale features is proposed, which can react to the anomalies in the process in a timely and effective manner to guarantee production safety. First, a graph-building method for spatio-temporally coupled time-series data using multidimensional time-varying feature map embedding is designed to capture the dependence of the data on time, while the topology of the graph is utilized to learn the spatial coupling of the data; second, a graph similarity-based anomaly monitoring strategy is innovatively proposed to measure the anomalies of the process using the difference degree index between the standard normal process data and the monitoring data. Finally, the proposed method is validated using the standard normal operating condition data of the Tennessee-Eastman (TE) process as well as the standard fault data. The experimental results show that the proposed model can identify anomalies more quickly and accurately than other typical methods, which significantly improves the reliability and timeliness of industrial process anomaly monitoring. Full article

Fractals are geometric patterns that appear self-similar across all length scales and are constructed by repeating a single unit on a regular basis. Entropy, as a core thermodynamic function, is an extension based on information theory (such as Shannon entropy) that is used to describe the topological structural complexity or degree of disorder in networks. Topological indices, as graph invariants, provide quantitative descriptors for characterizing global structural properties. In this paper, we investigate two types of generalized Sierpiński graphs constructed on the basis of different seed graphs, and employ six topological indices—the first Zagreb index, the second Zagreb index, the forgotten index, the augmented Zagreb index, the Sombor index, and the elliptic Sombor index—to analyze the corresponding entropy. We utilize the method of edge partition based on vertex degrees and derive analytical formulations for the first Zagreb entropy, the second Zagreb entropy, the forgotten entropy, the augmented Zagreb entropy, the Sombor entropy, and the elliptic Sombor entropy. This research approach, which integrates entropy with Sierpiński network characteristics, furnishes novel perspectives and instrumental tools for addressing challenges in chemical graph theory, computer networks, and other related fields. Full article

This paper develops a machine learning methodology for the rapid and robust prediction of the glass transition temperature (Tg) for polymers for the targeted application of sustainable high-temperature polymers. The machine learning framework combines multiple techniques to develop a feature set encompassing all relative aspects of polymer chemistry, to extract and explain correlations between features and Tg, and to develop and apply a high-throughput predictive model. In this work, we identify aspects of the chemistry that most impact Tg, including a parameter related to rotational degrees of freedom and a backbone index based on a steric hindrance parameter. Building on this scientific understanding, models are developed on different types of data to ensure robustness, and experimental validation is obtained through the testing of new polymer chemistry with remarkable Tg. The ability of our model to predict Tg shows that the relevant information is contained within the topological descriptors, while the requirement of non-linear manifold transformation of the data also shows that the relationships are complex and cannot be captured through traditional regression approaches. Building on the scientific understanding obtained from the correlation analyses, coupled with the model performance, it is shown that the rigidity and interaction dynamics of the polymer structure are key to tuning for achieving targeted performance. This work has implications for future rapid optimization of chemistries Full article