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The prevalent approach to motif analysis seeks to describe the local connectivity structure of networks by identifying subgraph patterns that appear significantly more often in a network then expected under a null model that conserves certain features of the original network. In this article we advocate for an alternative approach based on statistical inference of generative models where nodes are connected not only by edges but also copies of higher order subgraphs. These models naturally lead to the consideration of latent states that correspond to decompositions of networks into higher order interactions in the form of subgraphs that can have the topology of any simply connected motif. Being based on principles of parsimony the method can infer concise sets of motifs from within thousands of candidates allowing for consistent detection of larger motifs. The inferential approach yields not only a set of statistically significant higher order motifs but also an explicit decomposition of the network into these motifs, which opens new possibilities for the systematic study of the topological and dynamical implications of higher order connectivity structures in networks. After briefly reviewing core concepts and methods, we provide example applications to empirical data sets and discuss how the inferential approach addresses current problems in motif analysis and explore how concepts and methods common to motif analysis translate to the inferential framework. [ABSTRACT FROM AUTHOR]

Topological indices are mathematical descriptors of the structure of a molecule that can be used to predict its properties. They are derived from the graph theory, which describes the topology of a molecule and its connectivity. The main objective is mathematical modeling and topological properties of Y-graphyne. Current research focuses on two structures made from hexagonal honeycomb graphite lattices named triangular Y-graphyne and triangular Y-graphyne chains. The authors have simultaneously computed the first and second Reverse Zagreb indices, reverse hyper-Zagreb indices, and their polynomials. This research also derives mathematical closed-form formulas for some of its fundamental degree-based molecular descriptors. Researchers have been trying to synthesize a novel carbon form called Graphyne. For over a decade but with no success. Recently, some researchers have made a breakthrough in generating Carbons elusive allotrope and solved a long-standing problem in carbon materials. This wonder material is created to rival the conductivity of graphene but with control. These results opened new ways of research in the fields of semiconductors, electronics and optics. Furthermore, graphical and tabular results will help to investigate the structure-property relationships in γ-graphyne. [ABSTRACT FROM AUTHOR]

In the paper, we introduce the signless Laplacian A B C -matrix Q ̃ (G) = D ¯ (G) + A ̃ (G) , where D ¯ (G) is the diagonal matrix of A B C -degrees and A ̃ (G) is the A B C -matrix of G. The eigenvalues of the matrix Q ̃ (G) are the signless Laplacian A B C -eigenvalues of G. We give some basic properties of the matrix Q ̃ (G) , which includes relating independence number and clique number with signless Laplacian A B C -eigenvalues. For bipartite graphs, we show that the signless Laplacian A B C -spectrum and the Laplacian A B C -spectrum are the same. We characterize the graphs with exactly two distinct signless Laplacian A B C -eigenvalues. Also, we consider the problem of the characterization of the graphs with exactly three distinct signless Laplacian A B C -eigenvalues and solve it for bipartite graphs and, in some cases, for non-bipartite graphs. We also introduce the concept of the trace norm of the matrix Q ̃ (G) − t r (Q ̃ (G)) n I , called the signless Laplacian A B C -energy of G. We obtain some upper and lower bounds for signless Laplacian A B C -energy and characterize the extremal graphs attaining it. Further, for graphs of order at most 6, we compare the signless Laplacian energy and the A B C -energy with the signless Laplacian A B C -energy and found that the latter behaves well, as there is a single pair of graphs with the same signless Laplacian A B C -energy unlike the 26 pairs of graphs with same signless Laplacian energy and eight pairs of graphs with the same A B C -energy. [ABSTRACT FROM AUTHOR]

Let S(Γ) be a Seidel matrix of a graph Γ of order n and let D(Γ) = diag(n − 1 − 2d1, n − 1 − 2d2, ..., n − 1 − 2dn) be a diagonal matrix with di denoting the degree of a vertex vi in Γ. The Seidel Laplacian matrix of Γ is defined as SL(Γ) = D(Γ) − S(Γ). In this paper, we obtain an upper bound, and a lower bound on the Seidel Laplacian Estrada index of graphs. Moreover, we find a relation between Seidel energy and Seidel Laplacian energy of graphs. We establish some lower bounds on the Seidel Laplacian energy in terms of different graph parameters. Finally, we present a relation between Seidel Laplacian Estrada index and Seidel Laplacian energy of graphs. [ABSTRACT FROM AUTHOR]

Mycobacterium tuberculosis is an organism that causes tuberculosis (TB), a common infectious disease that has a high death and morbidity rate. Topological indices are mathematical tools used to describe the structural properties of molecules or networks. They provide a quantitative measure of the connectivity and complexity of a system, and play a crucial role in numerous area such as biochemistry and bioinformatics. The purpose of topological indices is to simplify complex structures into numerical values that can be easily analyzed and compared. QSPR modeling is a technique in chemistry that relates the structure of a chemical compound to its physical or chemical properties. It is used to predict properties like boiling points, solubilities, toxicities, and even biological activities of compounds. This saves time, resources, and enables researchers to make informed decisions in drug discovery, material science, and many other areas. In this study, we conducted an analysis of several drugs used for the treatment of tuberculosis. We focused on computing the reducible topological indices based on their degrees. Several techniques and approaches are employed. To perform calculations, we used edge partition methodology, analytical techniques, theoretical graph utilities, and degree counting method. Additionally, we examined six physicochemical properties of these drugs. To establish quantitative structure-property relationship models and evaluate their effectiveness, we employed linear, quadratic, and logarithmic regression analysis. By analyzing the reducible topological indices and physicochemical properties, we aimed to gain a deeper understanding of the drugs' characteristics and their potential impact on tuberculosis treatment. This study established a significant relationship between the defined indices with two key properties: molar mass and collision cross section. The correlation coefficients for molar mass range from 0.7 to 0.9, while the collision cross section range from 0.8 to 0.9. These results demonstrate a strong association between the indices and the properties under investigation. Furthermore, it is worth noting that both molar mass and collision cross section satisfy the requirements for p-value and F-test value across all indices. This indicates the statistical significance of the observed correlations and the reliability of our findings. [ABSTRACT FROM AUTHOR]

Previous studies on the co-evolving between vaccination strategies and epidemics mainly assumed that the vaccination strategies were made in the period between two spreading seasons. However, individual cognition during the spreading seasons might also alter the vaccination strategy and inversely influence the epidemic spreading. We propose a coupled disease-behavior model to describe the dynamic evolution of vaccination behavior during the spread of infectious diseases. The model integrates a susceptible-infected-vaccinated (SIV) model with the diffusion of vaccination behavior. We focus on the trade-off between perceptions of infection risk and the vaccination behaviors of neighbors, characterizing individual vaccination opinions. We introduce an opinion-critical value to map vaccination opinions into vaccination behavior. The vaccination coverage of the disease-behavior model is studied in network models and real-world networks. In addition, when societal costs are measured based on the degree of initial vaccinees, the cost of randomly selecting initial vaccinees is lowerthan selecting individuals with high or low degrees as vaccinees. Evaluating an individual's ability to transmit vaccination behavior based on the neighbor's number is inappropriate. We find that the impact of effective spreading rates on group vaccination is not one-sided and that reducing fear and highlighting the dangers of infectious diseases are crucial to increasing vaccination coverage. [ABSTRACT FROM AUTHOR]

Cubic fuzzy graphs (CFGs) offer greater utility as compared to interval-valued fuzzy graphs and fuzzy graphs due to their ability to represent the degree of membership for vertices and edges using both interval and fuzzy number forms. The significance of these concepts motivates us to analyze and interpret intricate networks, enabling more effective decision making and optimization in various domains, including transportation, social networks, trade networks, and communication systems. This paper introduces the concepts of vertex and edge connectivity in CFGs, along with discussions on partial cubic fuzzy cut nodes and partial cubic fuzzy edge cuts, and presents several related results with the help of some examples to enhance understanding. In addition, this paper introduces the idea of partial cubic α-strong and partial cubic δ-weak edges. An example is discussed to explain the motivation behind partial cubic α-strong edges. Moreover, it delves into the introduction of generalized vertex and edge connectivity in CFGs, along with generalized partial cubic fuzzy cut nodes and generalized partial cubic fuzzy edge cuts. Relevant results pertaining to these concepts are also discussed. As an application, the concept of generalized partial cubic fuzzy edge cuts is applied to identify regions that are most affected by trade deficits resulting from street crimes. Finally, the research findings are compared with the existing method to demonstrate their suitability and creativity. [ABSTRACT FROM AUTHOR]

This paper investigates the problem of resilient vector consensus for a group of dynamic agents against mobile malicious attacks. As real networks often operate under random environment and noises, we approach this problem by considering general random dynamic networks with weighted directed topologies. We propose three types of mobile attack models, which differ in the timing of moving of attackers and the capability of detecting such moving. By employing distributed discrete-time algorithms, the Lyapunov theory and martingale convergence theorem, resilient vector consensus is shown to be reached for all three models when the underlying network satisfies certain stochastic robustness conditions. [ABSTRACT FROM AUTHOR]