The topological structure of almost all interconnection networks is constructed in a recursive manner. For an [Formula: see text]-dimensional recursive network [Formula: see text] can be constructed by some different copies of [Formula: see text] by adding some edges among them. The reliability and stability of recursive networks are usually measured by the [Formula: see text]-embedded edge connectivity. For an [Formula: see text]-dimensional recursive network [Formula: see text], the [Formula: see text]-embedded edge connectivity of [Formula: see text], denoted by [Formula: see text], is the minimum size of edges whose removal disconnects [Formula: see text] and each vertex in the resulting network is in a [Formula: see text]-dimensional subnetwork [Formula: see text]. The [Formula: see text]-dimensional alternating group network [Formula: see text] is a typical recursive network. In this paper, we show that [Formula: see text] for [Formula: see text] and [Formula: see text].

As a pivotal research topic in the scheduling domain, the Flexible Job Shop Scheduling Problem (FJSP) has long attracted significant academic attention. However, research on FJSP involving transportation equipment capacity constraints remains relatively scarce. To address this gap, this paper proposes a Two-Layer Reinforcement Learning-Integrated Genetic Algorithm (TRLIGA) that simultaneously minimizes makespan and reduces energy consumption. The algorithmic framework integrates pre-scheduling with dynamic rescheduling constrained by transportation equipment capacity. This paper constructs a four-layer encoding structure and proposes a corresponding initialization method. During iteration, elite solutions are archived into an external non-dominated set for co-evolution, while a two-layer reinforcement learning framework adaptively tunes key parameters. A dedicated repair mechanism handles infeasible solutions. Experimental results demonstrate the effectiveness and superiority of the proposed methodology across comprehensive test scenarios.

Wireless Sensor Networks (WSNs) are emerging as a vital technology in future network utilization due to their broad application range and cost-effectiveness. The Exclusion Basis System (EBS) is a combinatorial technique of group key management that offers long-term and flexible security mechanism in WSNs while allowing the efficient eviction of compromised nodes and secure updates to the key system through its re-keying strategy. Combinatorial design theory is an innovative technique to developing and analysing cryptographic algorithms for secure communication in WSNs. This article generalised the study of N. Karst and S. B. Wicker to [Formula: see text]-design with [Formula: see text], triangular Partially Balanced Incomplete Block Designs (PBIBD) with [Formula: see text]=1 and [Formula: see text]=0, transversal design (TD[Formula: see text]), and [Formula: see text]-net. We observed that [Formula: see text]-design ([Formula: see text]) with [Formula: see text] provides less re-keying value than N. Karst and S. B. Wicker result [34]. Furthermore, when [Formula: see text], TD[Formula: see text] and triangular PBIBD yield the same re-keying values, whereas when [Formula: see text], TD[Formula: see text] provides a lower re-keying value than triangular PBIBD, except at [Formula: see text], where they are equal. Additionally, when [Formula: see text] and [Formula: see text] triangular PBIBD shows lower re-keying values than TD[Formula: see text] if [Formula: see text] otherwise equal and when [Formula: see text], the re-keying value of triangular PBIBD is consistently lower than TD[Formula: see text]. Finally, when [Formula: see text], TD[Formula: see text] and [Formula: see text]-net provide the same re-keying value.

We study approximation algorithms for the Min-Max Rural Postmen Cover Problem (MMRPCP). Given an undirected graph [Formula: see text], and a required subset [Formula: see text] of edges, where each edge in [Formula: see text] has a nonnegative weight, the objective is to find at most [Formula: see text] closed walks covering all the edges in [Formula: see text] such that the maximum weight of the closed walks is minimum. We propose a bicriteria [Formula: see text]-approximation algorithm for the MMRPCP. More exactly, given any instance [Formula: see text] of the MMRPCP consisting of a positive integer [Formula: see text], a graph [Formula: see text] and a required edge set [Formula: see text], the algorithm can produce at most [Formula: see text] closed walks covering all the edges in [Formula: see text] such that the maximum weight of the closed walks is no more than [Formula: see text] times the optimal value of [Formula: see text]. Previously, the best-known approximation ratio for the MMRPCP is [Formula: see text]. Our result demonstrates that a moderate relaxation of the constraint on the number of closed walks is helpful to reduce the approximation ratio.

Large scale multiprocessor systems based on interconnection networks (graphs) as underlying topologies, have been widely used in nowdays. Thus, the reliability of interconnection networks will directly affect the stability of multiprocessor systems. The strong Menger connectivity is a significant parameter for the reliability evalution of interconnection networks. A connected graph [Formula: see text] is strong Menger (edge) connected if there are [Formula: see text] internally (edge) disjoint paths joining any two vertices [Formula: see text] and [Formula: see text] in [Formula: see text]. This article studies the strong Menger connectivity of complete transposition networks [Formula: see text] when parts of subnetworks are faulty. Let [Formula: see text] be an [Formula: see text]-dimensional subnetwork of [Formula: see text]. We show that [Formula: see text] is strong Menger (edge) connected, and determine the (edge) connectivity of [Formula: see text] is [Formula: see text].

Embedding between two interconnection networks is a technique used to simulate and implement parallel algorithms in parallel processing and computing systems. For an embedding, edge congestion refers to the maximum count of edges of the guest graph being embedded into a single edge of the host graph. Wirelength or layout of an embedding is the sum of congestion on each edge of the host graph. Wirelength problem refers to finding the minimum possible wirelength between two structures of all possible embeddings. Minimizing the wirelength decreases wiring area which in turn reduces the cost and communication delay among the parallel processing components. Properties like regularity and the smaller number of inter-processor connections in hypercube and hypercube variants have made them prominent structures in the field of study and have been explored extensively. The half hypercube, constructed keeping hypercubes as fundamental blocks, exhibits several advantageous properties that are necessary for the effective selection of interconnection networks, such as reduced overhead, symmetry, fewer edges, and a smaller diameter. The paper aims to resolve the wirelength problem of embedding half hypercube to necklace and windmill graphs.

2-rotation symmetric Boolean functions have important applications in cryptography. This paper focuses on constructing 2-rotation symmetric Boolean functions having nice cryptographic properties. Firstly, we present some new constructions of 2-rotation symmetric bent functions with optimal algebraic degree. Secondly, we propose a class of 2-rotation symmetric resilient functions having high nonlinearity and optimal algebraic degree.

Early in 1992, Deutsch-Jozsa algorithm computed a symmetric partial Boolean function with a single quantum query, and thus achieved the best separation between classical deterministic and exact quantum query complexity. Recently, it was clarified that all symmetric partial Boolean functions with a single quantum query can be computed exactly by Deutsch-Jozsa algorithm. For the general partial Boolean functions with a single quantum query, the latest characterizations is complex and not very satisfactory. Based on this, this paper proves and discovers three new results: (1) Under a new equivalence, each partial Boolean function with a single quantum query can be transformed to a simple partial Boolean function whose polynomial degree is just one; (2) For partial Boolean functions up to four bits, there are only 10 non-trivial partial Boolean functions with a single quantum query; (3) For each quantum 1-query algorithm with undefined measurement, there exists a constructive method for finding out all partial Boolean functions that can be computed exactly by the algorithm.

Synchronised shuffle operators allow to specify symbols on which the operands must or can synchronise instead of interleave. Recently, partial derivative and position based automata for regular expressions with synchronised shuffle operators were introduced. In this paper, using the framework of analytic combinatorics, we study the asymptotic average state complexity of partial derivative automata for regular expressions with strongly and arbitrarily synchronised shuffles. The new results extend and improve the ones previously obtained for regular expressions with shuffle and intersection, as these operations can be seen as special cases of synchronised shuffles. For intersection, asymptotically the average state complexity of the partial derivative automaton is [Formula: see text], which significantly improves the known exponential upper-bound.