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The aim of this work is the reduction of the throttling noise when the capillary is used as a throttling device. Based on the theory of bubble dynamics, two-phase flow, and aerated supercavitation, four different sizes of aerated devices used in refrigerator refrigeration systems are designed. Throttling noise and the temperature and pressure of inlet and outlet of the capillary are measured under stable operation. To compare the noise suppression effects in different groups of experiments, we introduced the cavitation number to analyze, revealed the principle of aerated supercavitation to suppress noise, and combined the results of Fluent simulations to get the relationship between the noise suppression effect and the aerated quality. The experimental results showed that the aerated device can obviously suppress the throttling noise of the capillary outlet, up to 2.63 dB(A), which provides a new way for reducing the capillary throttling noise.

Let Hn be the linear heptagonal networks with 2n heptagons. We study the structure properties and the eigenvalues of the linear heptagonal networks. According to the Laplacian polynomial of Hn, we utilize the method of decompositions. Thus, the Laplacian spectrum of Hn is created by eigenvalues of a pair of matrices: LA and LS of order numbers 5n+1 and 4n+1n!/r!n−r!, respectively. On the basis of the roots and coefficients of their characteristic polynomials of LA and LS, we get not only the explicit forms of Kirchhoff index but also the corresponding total number of spanning trees of Hn.

Sierpinski networks are networks of fractal nature having several applications in computer science, music, chemistry, and mathematics. These networks are commonly used in chaos, fractals, recursive sequences, and complex systems. In this article, we compute various connectivity polynomials such as M-polynomial, Zagreb polynomials, and forgotten polynomial of generalized Sierpinski networks Skn and recover some well-known degree-based topological indices from these. We also compute the most general Zagreb index known as α,β-Zagreb index and several other general indices of similar nature for this network. Our results are the natural generalizations of already available results for particular classes of such type of networks.

This study provided a content analysis of studies aiming to disclose how artificial intelligence (AI) has been applied to the education sector and explore the potential research trends and challenges of AI in education. A total of 100 papers including 63 empirical papers (74 studies) and 37 analytic papers were selected from the education and educational research category of Social Sciences Citation Index database from 2010 to 2020. The content analysis showed that the research questions could be classified into development layer (classification, matching, recommendation, and deep learning), application layer (feedback, reasoning, and adaptive learning), and integration layer (affection computing, role-playing, immersive learning, and gamification). Moreover, four research trends, including Internet of Things, swarm intelligence, deep learning, and neuroscience, as well as an assessment of AI in education, were suggested for further investigation. However, we also proposed the challenges in education may be caused by AI with regard to inappropriate use of AI techniques, changing roles of teachers and students, as well as social and ethical issues. The results provide insights into an overview of the AI used for education domain, which helps to strengthen the theoretical foundation of AI in education and provides a promising channel for educators and AI engineers to carry out further collaborative research.

A graph’s entropy is a functional one, based on both the graph itself and the distribution of probability on its vertex set. In the theory of information, graph entropy has its origins. Hex-derived networks have a variety of important applications in medication store, hardware, and system administration. In this article, we discuss hex-derived network of type 1 and 2, written as HDN 1n and HDN 2n, respectively of order n. We also compute some degree-based entropies such as Randić, ABC, and GA entropy of HDN 1n and HDN 2n.

A connected graph is called Hamilton-connected if there exists a Hamiltonian path between any pair of its vertices. Determining whether a graph is Hamilton-connected is an NP-complete problem. Hamiltonian and Hamilton-connected graphs have diverse applications in computer science and electrical engineering. The detour index of a graph is defined to be the sum of lengths of detours between all the unordered pairs of vertices. The detour index has diverse applications in chemistry. Computing the detour index for a graph is also an NP-complete problem. In this paper, we study the Hamilton-connectivity of convex polytopes. We construct three infinite families of convex polytopes and show that they are Hamilton-connected. An infinite family of non-Hamilton-connected convex polytopes is also constructed, which, in turn, shows that not all convex polytopes are Hamilton-connected. By using Hamilton connectivity of these families of graphs, we compute exact analytical formulas of their detour index.

In this study, we first introduce polygonal cylinder and torus using Cartesian products and topologically identifications and then find their Wiener and hyper-Wiener indices using a quick, interesting technique of counting. Our suggested mathematical structures could be of potential interests in representation of computer networks and enhancing lattice hardware security.

A complete classification of simple function germs with respect to Lipschitz equivalence over the field of complex numbers ℂ was given by Nguyen et al. The aim of this article is to implement a classifier in terms of easy computable invariants to compute the type of the Lipschitz simple function germs without computing the normal form in the computer algebra system Singular.

Let G=VG,EG be the connected graph. For any vertex i∈VG and a subset B⊆VG, the distance between i and B is di;B=mindi,j|j∈B. The ordered k-partition of VG is Π=B1,B2,…,Bk. The representation of vertex i with respect to Π is the k-vector, that is, ri|Π=di,B1,di,B2,…,di,Bk. The partition Π is called the resolving (distinguishing) partition if ri|Π≠rj|Π, for all distinct i,j∈VG. The minimum cardinality of the resolving partition is called the partition dimension, denoted as pdG. In this paper, we consider the upper bound for the partition dimension of the generalized Petersen graph in terms of the cardinalities of its partite sets.

After the occurrence of public health emergencies, due to the uncertainty of the evolution of events and the asymmetry of pandemic information, the public’s risk perception will fluctuate dramatically. Excessive risk perception often causes the public to overreact to emergencies, resulting in irrational behaviors, which have a negative impact on economic development and social order. However, low-risk perception will reduce individual awareness of prevention and control, which is not conducive to the implementation of government pandemic prevention and control measures. Therefore, it is of great significance to accurately evaluate public risk perception for improving government risk management. This paper took the evolution of public risk perception based on the COVID-19 region as the research object. First, we analyze the characteristics of infectious diseases in the evolution of public risk perception of public health emergencies. Second, we analyze the characteristics of risk perception transmission in social networks. Third, we establish the dynamic model of public risk perception evolution based on SEIR, and the evolution mechanism of the public risk perception network is revealed through simulation experiments. Finally, we provide policy suggestions for government departments to deal with public health emergencies based on the conclusions of this study.

In this work, we introduce a new topological index called a general power sum-connectivity index and we discuss this graph invariant for some classes of extremal graphs. This index is defined by YαG=∑uv∈EGdudu+dvdvα, where du and dv represent the degree of vertices u and v, respectively, and α≥1. A connected graph G is called a k-generalized quasi-tree if there exists a subset Vk⊂VG of cardinality k such that the graph G−Vk is a tree but for any subset Vk−1⊂VG of cardinality k−1, the graph G−Vk−1 is not a tree. In this work, we find a sharp lower and some sharp upper bounds for this new sum-connectivity index.

This paper focuses on the comparative study of natural convection flow of fractional Maxwell fluid having uniform heat flux and radiation. The well-known Maxwell fluid equation with an integer-order derivative has been extended to a non-integer-order derivative, i.e., fractional derivative. The explicit expression for the temperature and velocity is acquired by utilizing the Laplace transform (LT) technique. The two fractional derivative concepts are used (Caputo and Caputo–Fabrizio derivatives) in the formulation of the problem. Utilizing the Mathcad programming, the effect of certain embedded factors and fractional parameters on temperature and velocity profile is graphically presented.

In this paper, we give the relation between the spectrum of strongly regular graph and its clique-inserted graph. The Laplacian spectrum and the signless Laplacian spectrum of clique-inserted graph of strongly regular graph are calculated. We also give formulae expressing the energy, Kirchoff index, and the number of spanning trees of clique-inserted graph of a strongly regular graph. And, clique-inserted graph of the triangular graph Tt, which is a strongly regular graph, is enumerated.

A graph’s entropy is a functional one, based on both the graph itself and the distribution of probability on its vertex set. In the theory of information, graph entropy has its origins. Dominating David derived networks have a variety of important applications in medication store, hardware, and system administration. In this study, we discuss dominating David derived network of type 1, 2, and 3 written as D1n, D2n, and D3n, respectively of order n. We also compute some degree-based entropies such as Randić, ABC, and GA entropy of D1n, D2n, and D3n.

The study of structure-property relations including the transformations of molecules is of utmost importance in correlations with corresponding physicochemical properties. The graph topological indices have been used effectively for such study and, in particular, bond-based indices play a vital role. The bond-additive topological indices of a molecular graph are defined as a sum of edge measures over all edges in which edge measures can be computed based on degrees, closeness, peripherality, and irregularity. In this study, we provide the mathematical characterization of the transformation of a structure that can be accomplished by the novel edge adjacency and incidence relations. We derive the exact expressions of bond type indices such as second Zagreb, sigma indices, and their coindices of total transformation and two types of semitransformations of the molecules which in turn can be used to characterize the topochemical and topostructural properties.

Let Γ be a simple connected undirected graph with vertex set VΓ and edge set EΓ. The metric dimension of a graph Γ is the least number of vertices in a set with the property that the list of distances from any vertex to those in the set uniquely identifies that vertex. For an ordered subset W=w1,w2,…,wk of vertices in a graph Γ and a vertex v of Γ, the metric representation of v with respect to W is the k-vector rvW=dv,w1,dv,w2,…,dv,wk. If every pair of distinct vertices of Γ have different metric representations, then the ordered set W is called a resolving set of Γ. It is known that the problem of computing this invariant is NP-hard. In this paper, we consider the problem of determining the cardinality ψΓ of minimal doubly resolving sets of Γ and the strong metric dimension for the jellyfish graph JFGn,m and the cocktail party graph CPk+1.

Computing the resistance distance of a network is a fundamental and classical topic. In the aspects of considering the resistances between any two points of the lattice networks, there are many studies associated with the ladder networks and ladderlike networks. But the resistances between any two points for more complex structures than ladder networks or ladderlike networks are still unknown. In this paper, a rather complicated structure which is named linear octagonal network is considered. Treelike octagonal systems are cata-condensed systems of octagons, which represent a class of polycyclic conjugated hydrocarbons. A linear octagonal network is a cata-condensed octagonal system with no branchings. Moreover, the resistances between any two points of a linear octagonal network are first determined. One finds that the effective resistances between new inserted points and others points of a linear octagonal network can be given by the effective resistances between two initial points which are inherited from the linear polyomino network.

Let G be a finite, connected graph of order of, at least, 2 with vertex set VG and edge set EG. A set S of vertices of the graph G is a doubly resolving set for G if every two distinct vertices of G are doubly resolved by some two vertices of S. The minimal doubly resolving set of vertices of graph G is a doubly resolving set with minimum cardinality and is denoted by ψG. In this paper, first, we construct a class of graphs of order 2n+Σr=1k−2nmr, denoted by LSGn,m,k, and call these graphs as the layer Sun graphs with parameters n, m, and k. Moreover, we compute minimal doubly resolving sets and the strong metric dimension of the layer Sun graph LSGn,m,k and the line graph of the layer Sun graph LSGn,m,k.

The hypercube Qn is one of the most admirable and efficient interconnection network due to its excellent performance for some practical applications. The Kirchhoff index KfG is equal to the sum of resistance distances between any pairs of vertices in networks. In this paper, we deduce some bounds with respect to Kirchhoff index of hypercube network Qn.

In mathematics, one always tries to get new structures from given ones. This also applies to the realm of graphs, where one can generate many new graphs from a given set of graphs. In this work, using knowledge of difference equations, we drive the explicit formulas for the number of spanning trees in the sequence of some graphs generated by a triangle by electrically equivalent transformations and rules of weighted generating function. Finally, we compare the entropy of our graphs with other studied graphs with average degree being 4, 5, and 6.

The long-periodic/infinite discrete Gabor transform (DGT) is more effective than the periodic/finite one in many applications. In this paper, a fast and effective approach is presented to efficiently compute the Gabor analysis window for arbitrary given synthesis window in DGT of long-periodic/infinite sequences, in which the new orthogonality constraint between analysis window and synthesis window in DGT for long-periodic/infinite sequences is derived and proved to be equivalent to the completeness condition of the long-periodic/infinite DGT. By using the property of delta function, the original orthogonality can be expressed as a certain number of linear equation sets in both the critical sampling case and the oversampling case, which can be fast and efficiently calculated by fast discrete Fourier transform (FFT). The computational complexity of the proposed approach is analyzed and compared with that of the existing canonical algorithms. The numerical results indicate that the proposed approach is efficient and fast for computing Gabor analysis window in both the critical sampling case and the oversampling case in comparison to existing algorithms.

We introduce an informative labeling algorithm for the vertices of a family of Koch networks. Each label consists of two parts: the precise position and the time adding to Koch networks. The shortest path routing between any two vertices is determined only on the basis of their labels, and the routing is calculated only by few computations. The rigorous solutions of betweenness centrality for every node and edge are also derived by the help of their labels. Furthermore, the community structure in Koch networks is studied by the current and voltage characterizations of its resistor networks.