The strong product serves as an essential method to build parallel processing network models utilizing a number of small graphs. The network models constructed through the strong product incorporate these small graphs as subgraphs and preserve many of the advantageous properties of the factor graphs. The [Formula: see text]-distant Hamiltonian walk indicates a generalization of the Hamiltonian cycle, and the [Formula: see text]-distant Hamiltonian walk in the graph demonstrates a cyclic sequence of all its vertices, where the distance between two consecutive vertices is [Formula: see text]. In the design of wireless sensor networks, the [Formula: see text]-distant Hamiltonian walk plays an important role. In this paper, sufficient conditions are determined for the existence of [Formula: see text]-distant Hamiltonian walks in the strong product of simple, connected, undirected graphs. These conditions are derived on the basis of the connectivity, degree, and specific edge connectivity of the graph. The existence of [Formula: see text]-distant Hamiltonian walks is tested by exploring the strong product topology, with relevant theorems and examples provided, and corresponding algorithms given to verify the applicability and effectiveness of the parallel network model proposed in this paper.

A barrel shifter is a digital circuit used to shift the data word by predetermining the number of bits in a single clock. Typically, the multiplexers are connected in sequence to perform shifting operations. The output of one multiplexer is connected to the input of another multiplexer according to the chosen shift distance. The total number of multiplexers required for implementation is determined by equation [Formula: see text]. In this paper, an area-minimised barrel shifter is proposed, which reduces the number of multiplexers by [Formula: see text]. These reduced multiplexers are replaced by a NOT-AND gate. The proposed design is modelled using Verilog and synthesised in the Xilinx Vivado suite. MOS transistor implementation is verified in Microwind and DSCH tool. Experimental results show that the design is properly working and reduces the transistor count by 19.17%. The proposed design is best suited for area-constraint implementations.

Diagnosis plays an important role in measuring the reliability of interconnection networks, and the diagnosability of interconnection networks has been widely investigated. In 2012, Peng et al. proposed the g-good-neighbor diagnosability, which requires that every fault-free processor contains at least g fault-free neighbors. In this paper, we show that the g-good-neighbor diagnosability of folded hypercube-like networks under the PMC model and MM* model when [Formula: see text].

We study the possibility of designing [Formula: see text]-round protocols for problems of substantially super-linear polynomial-time (sequential) complexity on the congested clique with about [Formula: see text] nodes, where [Formula: see text] is the input size. We show that the average time complexity of the local computation performed at a clique node (in terms of the size of the data received by the node) in such protocols has to be substantially larger than the time complexity of the given problem.

Considering the mutual influence of product prices in adjacent regions on demand, in this paper, we develop a differential game model of transboundary pollution that incorporates emission reduction technologies. By applying optimal control theory, we derive the optimal strategies and pollution stock emission paths for two regions in both non-cooperative and cooperative game scenarios. A comparative analysis of the optimal results and an examination of parameter impacts are conducted for these two scenarios. The research results demonstrate that under cooperative game theory, product prices are lower, optimal production is higher, pollution levels in the air are lower, and social welfare is higher. Furthermore, as price competition intensifies, product prices exhibit a downward trend, while pollution stocks have increased accordingly. Conversely, advancements in emission reduction technology lead to increased regional benefits and a corresponding decrease in pollution stocks.

Connectivity is an important metric for fault tolerance in interconnection networks. Menger’s theorem reveals the relationship between connectivity and disjoint paths in a graph. Disjoint paths not only avoid communication bottlenecks, but also provide alternative paths in case of vertex failures. Let [Formula: see text] [Formula: see text] be a [Formula: see text]-dimensional sub-bubble-sort star of [Formula: see text]. In this paper, we show that [Formula: see text] is strongly Menger (edge) connected. Later, we show that the connectivity and edge-connectivity of [Formula: see text] are uniformly [Formula: see text]. In addition, we show that the 1-extra connectivity of [Formula: see text] is [Formula: see text] for [Formula: see text].

Cyclic codes are a subclass of linear codes. They have efficient encoding and decoding algorithms, so they are used for consumer electronics, data storage systems,and communication systems. In this paper, a class of three-weight cyclic codes over GF(2) whose duals have two zeros is presented. The weight distribution of this class of cyclic codes is settled by quadratic form theory.

Partial differential equations and systems with certain boundary conditions specify continuous processes significant for both large-scale simulations in computer-aided design using HPC and subsequent real-time control of embedded applications using dedicated hardware. The paper develops a spectrum of techniques based on a family of place-transition nets aimed at the computing and communication structure design for fast mass-parallel numerical solving of PDEs. For the HPC domain, we develop models of interconnects in the form of infinite nets and graphical programs in the form of Sleptsov nets. For the embedded control domain, we develop specialized lattices for fast numerical solving PDE based on integer number approximation specified with Sleptsov-Salwicki nets to be implemented on dedicated hardware, which we prototype on FPGAs. For mass-parallel solving of PDEs, we employ ad-hoc finite-difference schemes and iteration methods that allow us to recalculate the lattice values in a single time cycle suitable for control of hypersonic objects and thermonuclear reactions.

Diagnosability of a multiprocessor system is an important measure of the reliability of interconnection networks. In order to better measure the reliability of the system, Yuan et al. introduced the non-inclusive [Formula: see text]-good-neighbor diagnosability, denoted as [Formula: see text], which requires every pair of [Formula: see text]-good-neighbor faulty sets is non-inclusive and every fault-free vertex has at least [Formula: see text] fault-free neighbors. In the paper, we obtain the upper bound of the non-inclusive [Formula: see text]-good-neighbor diagnosability of star graphs, [Formula: see text] for [Formula: see text] under the [Formula: see text] model and the [Formula: see text] model, and determine the non-inclusive 1-good-neighbor diagnosability of star graphs, [Formula: see text] for [Formula: see text] under the [Formula: see text] model, [Formula: see text] for [Formula: see text] under the [Formula: see text] model.

The Hamiltonian problem is a fundamental research topic in graph theory. In this paper, we study a relaxation of the Hamiltonian problem. A graph [Formula: see text] is spanning [Formula: see text]-edge-cyclable if, for any [Formula: see text] independent edges [Formula: see text] of [Formula: see text], there exist [Formula: see text] vertex-disjoint cycles [Formula: see text] in [Formula: see text] such that [Formula: see text] and [Formula: see text] for all [Formula: see text]. Clearly, a graph [Formula: see text] is Hamiltonian if it is spanning [Formula: see text]-edge-cyclable. We focus on spanning [Formula: see text]-edge-cyclability of enhanced hypercube network, which is an important variant of well known hypercube. We prove that the [Formula: see text]-dimensional enhanced hypercube [Formula: see text] is spanning [Formula: see text]-edge-cyclable for [Formula: see text], [Formula: see text] and odd [Formula: see text].