Search Results (34)

Let us consider the finite commutative ring R, whose unity is $1\ne 0.$ Its weakly zero-divisor graph, represented as $W\Gamma \left(R\right)$, is a basic undirected graph with two distinct vertices, $c_1$ and $c_2$, that are adjacent if and only if there exist $r\in$ $\mathrm{ann}\left(c_1\right)$ and $s\in$ $\mathrm{ann}\left(c_2\right)$ that satisfy the condition $rs=0$. Let $D\left(G\right)$ be the distance matrix and $Tr\left(G\right)$ be the diagonal matrix of the vertex transmissions in basic undirected connected graph G. The $D_{\alpha }$ matrix of graph G is defined as $D_{\alpha }\left(G\right)=\alpha Tr\left(G\right)+(1-\alpha )D\left(G\right)$ for $\alpha \in [0,1]$. This article finds the $D_{\alpha }$ spectrum for the graph $W\Gamma \left({\mathbb{Z}}_n\right)$ for various values of n and also shows that $W\Gamma \left({\mathbb{Z}}_n\right)$ for $n={\vartheta }_1{\vartheta }_2{\vartheta }_3\dots {\vartheta }_t{\eta }_1^{d_1}{\eta }_2^{d_2}\dots {\eta }_s^{d_s}(d_i\ge 2,t\ge 1,s\ge 0)$, where ${\vartheta }_i$’s and ${\eta }_i$’s are the distinct primes, is $D_{\alpha }$ integral. Full article

The utilisation of 3D models in low-cost devices, such as the internet of things, virtual reality, and augmented reality, is expanding. The challenge lies in the lack of lightweight solutions for verifying the authenticity of models in the graphics library transmission format (glTF) on devices with limited resources. The glTF standard, which allows storage in glb format, is the leading standard for representing 3D assets. Despite its popularity, research on watermarking glTF models remains limited. This paper proposes a novel method for authenticating 3D models in glb format based on fragile watermarking. Additionally, an analysis was conducted to determine the impact of embedding the watermark in vertex attributes other than position on the integrity and visual quality of the model. The methodology is as follows: (1) embedding the watermark, (2) applying model modification or omitting it, and (3) verifying authenticity based on the recovered watermark. The proposed algorithm attaches a 512-bit hash-based message authentication code (HMAC) to a 3D model using the least significant bits (LSBs) modification method. The use of HMAC and LSBs has resulted in a computationally efficient algorithm that can be implemented in low-cost devices. Full article

Waves on graphs are a current subject of research interest. As opposed to flows on graphs, the reflection–transmission of waves at the graph’s vertex is a problem that needs to be further modelled mathematically. The literature on the reflection and transmission of waves at a vertex is scarce. Some articles are reviewed and discussed. Water waves are a good topic for comparing different mathematical models, from hyperbolic conservation laws to weakly nonlinear, weakly dispersive systems of partial differential equations on a two-dimensional fattened (thick) graph and the respective one-dimensional graph-model reduction. In this study, we present a particular water wave model in which junction angles are systematically included in the mathematical model. Comparing the solutions with the fattened-graph model gave rise to a more general compatibility condition at the vertex. Current research topics of interest are outlined at the end. Full article

Recently, there has been active research on utilizing GPUs for the efficient processing of large-scale dynamic graphs. However, challenges arise due to the repeated transmission and processing of identical data during dynamic graph operations. This paper proposes an efficient processing scheme for large-scale dynamic graphs in GPU environments with limited memory, leveraging dynamic scheduling and operation reduction. The proposed scheme partitions the dynamic graph and schedules each partition based on active and tentative active vertices, optimizing GPU utilization. Additionally, snapshots are employed to capture graph changes, enabling the detection of redundant edge and vertex modifications. This reduces unnecessary computations, thereby minimizing GPU workloads and data transmission costs. The scheme significantly enhances performance by eliminating redundant operations on the same edges or vertices. Performance evaluations demonstrate an average improvement of 280% over existing static graph processing techniques and 108% over existing dynamic graph processing schemes. Full article

Concatemeric viral DNA is packaged into bacteriophage P22 procapsids via a headful packaging mechanism mediated by a molecular machine consisting of small (gp3) and large (gp2) terminase subunits. Although a negative stain reconstruction exists for the terminase holoenzyme, it is not clear how this complex binds the dodecameric portal protein located at a 5-fold mismatch vertex. Herein, we describe new assemblies for the holoenzyme. Both native mass spectrometry and transmission electron microscopy reveal that the P22 terminase complex adopts three main assemblies, which include a nonameric S-terminase bound to two L-terminase 1(gp3)₉:2(gp2), two nonameric S-terminase bound to five L-terminase 2(gp3)₉:5(gp2), and three nonameric S-terminase bound to seven L-terminase 3(gp3)₉:7(gp2). Native agarose gel electrophoresis shows that the terminase complex interacts with procapsids with mild crosslinking. These results herein illustrate the P22 terminase complex can adopt a variety of conformations and assembly states. Full article

The status (or transmission) of a vertex in a connected graph is the sum of distances between the vertex and all other vertices. The minimum status (or minimum transmission) of a connected graph is the minimum of the statuses of all vertices in the graph. Previously, sharp lower and upper bounds have been obtained on the minimum status of connected graphs with a fixed maximum degree k and order n. Moreover, for $2\le k\le {\displaystyle \frac{n}{2}}$, the following theorem about graphs attaining the maximum on the minimum status has also been proposed without proof. The theorem is as follows: Let G be a connected graph of order n with $▵(G)=k$, where $2\le k\le {\displaystyle \frac{n}{2}}$. Then, the minimum status of G attains the maximum if and only if one of the following holds. (1) G is a path or a cycle, where $k=2$; (2) $G_{k,n}$ is a spanning subgraph of G and G is a spanning subgraph of $H_{k,n}$, where $3\le k<{\displaystyle \frac{n}{2}}$; and (3) either $G_{{\displaystyle \frac{n}{2}},n}$ is a spanning subgraph of G and G is a spanning subgraph of $H_{{\displaystyle \frac{n}{2}},n}$ or $G_{{\displaystyle \frac{n}{2}},n}$ is a spanning subgraph of G and G is a spanning subgraph of $H_n$, where $k={\displaystyle \frac{n}{2}}$ for even $n\ge 6$. For the integers $n,k$ with $2\le k\le n-1$, the graph $G_{k,n}$ has the vertex set $V(G_{k,n})=\{x_1,x_2,\cdots ,x_n\}$ and the edge set $E(G_{k,n})=\{x_i{x}_{i+1}:i=1,2,\cdots ,n-k\}\cup \{x_{n-k+1}{x}_j:j=n-k+2,n-k+3,\cdots ,n\}$; the graph $H_{k,n}$ is obtained from $G_{k,n}$ by adding all the edges $x_i{x}_j$, where $n-k+2\le i<j\le n$; and for even $n\ge 6$ the graph $H_n$ is obtained from $G_{{\displaystyle \frac{n}{2}},n}$ by adding the edge $x_{{\displaystyle \frac{n}{2}}-1}{x}_{{\displaystyle \frac{n}{2}}+2}$ and all the edges $x_i{x}_j$, where ${\displaystyle \frac{n}{2}}+3\le i<j\le n$. This study provides the proof to complete the above theorem. Full article

Against the background of the construction of new power systems, power generation, transmission, distribution, and dispatching services are open to the outside world for interaction, and the accessibility of attack paths has been significantly enhanced. We are facing cyber-physical cross-domain attacks with the characteristics of strong targeting, high concealment, and cross-space threats. This paper proposes a quantitative analysis method for the influence of power cyber-physical cross-domain attack paths based on graph knowledge. First, a layered attack graph was constructed based on the cross-space and strong coupling characteristics of the power cyber-physical system business and the vertical architecture of network security protection focusing on border protection. The attack graph included cyber-physical cross-domain attacks, control master stations, measurement and control equipment failures, transient stable node disturbances, and other vertices, and achieved a comprehensive depiction of the attack path. Second, the out-degree, in-degree, vertex betweenness, etc., of each vertex in the attack graph were comprehensively considered to calculate the vertex vulnerability, and by defining the cyber-physical coupling degree and edge weights, the risk of each attack path was analyzed in detail. Finally, the IEEE RTS79 and RTS96 node systems were selected, and the impact of risk conduction on the cascading failures of the physical space system under typical attack paths was analyzed using examples, verifying the effectiveness of the proposed method. Full article

As the core device of engine-driven vehicle powertrains, the automatic transmission (AT) can obtain different output torques and speeds. Designing new mechanisms is the foundation for developing innovative ATs. This paper proposes an innovative design method for 7-speed, 8-speed and 9-speed AT mechanisms based on fractionated planetary gear trains (PGTs) and investigates the performance analysis of the AT mechanisms. Firstly, the detection of symmetric vertices in the PGT graph is studied, and a method for the topologic synthesis of 11-link 2-degree-of-freedom (DOF) fractionated PGTs is proposed. Secondly, the innovative design of 7-speed, 8-speed and 9-speed AT mechanisms is explored, leading to the discovery of two novel 7-speed AT mechanisms, three novel 8-speed AT mechanisms and four novel 9-speed AT mechanisms. Finally, the performance analyses of kinematics, torque, power flow and transmission efficiency of the AT mechanisms are conducted, and AMESim (Simcenter Amesim 2020.1) software is utilized to simulate speed ratios and transmission efficiencies of the AT mechanisms. The analysis and simulation results demonstrate that the nine novel AT mechanisms obtained in this study exhibit excellent overall performances. The speed ratios of each AT mechanism are reasonable, the step ratio is between 1.1 and 1.8, and the transmission efficiency exceeds 0.95. There is no cyclic power, or the cyclic power values are within an acceptable limit. Full article

First, this study proposed a metamaterial beam model with spatially varying interval density. The interval dynamic equation of this model could be established by incorporating the decomposition results of the interval field based on Karhunen–Loeve expansion into the finite element method. An interval perturbation finite element method was developed to evaluate the bounds of the dynamic response interval vector. Then, an interval vibration transmission analysis could be performed, and the frequency range of the safe band gap could be determined. Meanwhile, Monte Carlo simulations and the vertex method are also presented to provide reference solutions. By comparison, it was found that the calculation accuracy of the interval perturbation finite element method was acceptable. The numerical results also showed that the safe band gap range was significantly smaller than that of the deterministic band gap. Full article

Modeling data transmission problems in graph theory is internalized to the existence of fractional flows, and thus can be surrogated to be characterized by a fractional factor in diversified settings. We study the fractional factor framework in the network environment when some sites are damaged. The setting we focus on refers to the lower and upper fractional degrees described by two functions on the vertex set. It is determined that G is fractional $(g,f,n)$ critical if $\delta (G)\ge \lfloor \frac{a^2+b^2+2ab+2a+2b-3}{4a}\rfloor +n$ and $I(G)>\frac{n+\lfloor \frac{{(a+b-1)}^2}{2a}+\frac{2b-1}{a}\rfloor }{2}$, where $1\le a\le b$ and $b\ge 2$. Full article

Air cycle systems (ACSs) are primarily used in aircraft environmental control systems (ECSs) to provide a suitable cabin temperature and pressure environment for passengers and avionics. It comprises heat exchangers, compressors, turbines, water separators, and various other components that are interconnected to form an information-transmission network. Traditional research on ACSs has focused primarily on their thermal performance. This study abstracted ACSs into network graphs based on their information-transmission characteristics, determined the weight of each information-transmission route using the fuel weight penalty method, calculated and compared the order degree of different ACSs using the structure entropy method, and measured the importance of each component using centrality for the first time. The results showed that the order degree of the ACSs gradually increased with an increase in the number of wheels in the air cycle machine (ACM), and ACSs with high-pressure water separation had a higher order degree under wet conditions than under dry conditions. Moreover, based on the centrality of each vertex in the graphs, the ACM and secondary heat exchanger in the ACS were fundamentally important and should be focused on during the system design. The methodology proposed in this study provides a theoretical basis for the evaluation of the ACS organizational structure and the design performance of components. Full article

In order to abstract away a network model from some real-world networks, such as navigation satellite networks and mobile call networks, we proposed an Isochronal-Evolution Random Matching Network (IERMN) model. An IERMN is a dynamic network that evolves isochronally and has a collection of edges that are pairwise disjoint at any point in time. We then investigated the traffic dynamics in IERMNs whose main research topic is packet transmission. When a vertex of an IERMN plans a path for a packet, it is permitted to delay the sending of the packet to make the path shorter. We designed a routing decision-making algorithm for vertices based on replanning. Since the IERMN has a specific topology, we developed two suitable routing strategies: the Least Delay Path with Minimum Hop (LDPMH) routing strategy and the Least Hop Path with Minimum Delay (LHPMD) routing strategy. An LDPMH is planned by a binary search tree and an LHPMD is planned by an ordered tree. The simulation results show that the LHPMD routing strategy outperformed the LDPMH routing strategy in terms of the critical packet generation rate, number of delivered packets, packet delivery ratio, and average posterior path lengths. Full article

Let G be a strongly connected digraph, and $d_G(v_i,v_j)$ denote the distance from the vertex $v_i$ to vertex $v_j$ and be defined as the length of the shortest directed path from $v_i$ to $v_j$ in G. The sum distance between vertices $v_i$ and $v_j$ in G is defined as $s{d}_G(v_i,v_j)=d_G(v_i,v_j)+d_G(v_j,v_i)$. The sum distance matrix of G is the $n\times n$ matrix $SD(G)={(s{d}_G(v_i,v_j))}_{v_i,v_j\in V(G)}$. For vertex $v_i\in V(G)$, the sum transmission of $v_i$ in G, denoted by $S{T}_G(v_i)$ or $S{T}_i$, is the row sum of the sum distance matrix $SD(G)$ corresponding to vertex $v_i$. Let $ST(G)=\mathrm{diag}(S{T}_1,S{T}_2,\dots ,S{T}_n)$ be the diagonal matrix with the vertex sum transmissions of G in the diagonal and zeroes elsewhere. For any real number $0\le \alpha \le 1$, the general sum distance matrix of G is defined as $S{D}_{\alpha }(G)=\alpha ST(G)+(1-\alpha )SD(G).$ The eigenvalues of $S{D}_{\alpha }(G)$ are called the general sum distance eigenvalues of G, the spectral radius of $S{D}_{\alpha }(G)$, i.e., the largest eigenvalue of $S{D}_{\alpha }(G)$, is called the general sum distance spectral radius of G, denoted by ${\mu }^{\alpha }(G)$. In this paper, we first give some spectral properties of $S{D}_{\alpha }(G)$. We also characterize the digraph minimizes the general sum distance spectral radius among all strongly connected r-partite digraphs. Moreover, for digraphs that are not sum transmission regular, we give a lower bound on the difference between the maximum vertex sum transmission and the general sum distance spectral radius. Full article

The transmission of a vertex v of a graph G is the sum of distances from v to all the other vertices of G. A transmission irregular graph (TI graph) has mutually distinct vertex transmissions. In 2018, Alizadeh and Klavžar posed the following question: do there exist infinite families of regular TI graphs? An infinite family of TI cubic graphs of order $118+72k$, $k\ge 0$, was constructed by Dobrynin in 2019. In this paper, we study the problem of finding TI cubic graphs for an arbitrary number of vertices. It is shown that there exists a TI cubic graph of an arbitrary even order $n\ge 22$. Almost all constructed graphs are contained in twelve infinite families. Full article

This paper aims to develop a method that could serve as a tool for evaluating extracted raw materials in terms of use by considering the place of extraction and consumption. Dijkstra´s algorithm solves many of the shortest path problems observed in the production planning of raw materials. The algorithm requires knowledge of the relative distance between the vertices and the definition of the Euclidean distance of the vertices from the target vertex. The algorithm scans all of the paths and chooses the one with the minimum distance. At the same time, it would be able to identify the places of sale of raw materials and transport sites for the transportation of raw materials. It would have a database of point and line sources of occurrence (mining, deposit), places of transport (transmission network), and points of sale (seller). At present, geo-statistics is becoming an essential tool for solving various problems in modern deposit geology. Its results are used to calculate reserves and the economic valuation of the deposit. In the process of production planning, it is necessary to constantly process and analyze the geological information obtained during the mining survey. Full article