The h-connectivity is a crucial metric for evaluating the fault tolerance of interconnection networks, as it offers a more precise and realistic assessment of performance under faulty conditions. The incremental scalability exchanged hypercube, denoted by ISEH(s,t,alpha) , is a recently proposed variant of the hypercube. It inherits advantageous properties from EH(s,t) , such as low complexity and high fault tolerance, while providing superior incremental scalability. In this work, we determine the edge connectivity and h-connectivity of ISEH(s,t,alpha) . These results provide a refined measure of the network ' s resilience and confirm its exceptional fault tolerance, offering a solid foundation for practical deployment in large-scale parallel computing systems.
Let p >= 5 be a prime and let R=Fpm+uFpm , where u2=0 . This paper classifies all self-dual lambda-constacyclic codes of length 3ps over R , where lambda is a unit. We distinguish two cases. If lambda not equal lambda-1 , we show that the unique self-dual lambda-constacyclic code is < u > . If lambda=lambda-1 , then lambda=+/- 1 . When pm equivalent to 1 (mod 3), the codes are obtained from those of length rho s . When pm equivalent to 2 (mod 3), we give an explicit structural characterization of all such codes.
This paper presents a novel transposition-based sorting algorithm, which integrates two distinct sorting paradigms, namely Insertion-Sort() and Merge-Sort(). The resulting algorithm, which we call Inserge-Sort(), exhibits several intriguing properties, which are rigorously demonstrated in the paper. Notably, our approach enables fine-grained control over the number of inversion pairs in the input array. A simple modification of our algorithm allows for an exact reduction in the number of inversion pairs. It remains to be determined whether other sorting algorithms can be suitably adapted for this purpose. Our algorithm can be thought of as rewriting Insertion-Sort(). The advantage of a rewrite algorithm is that it facilitates the justification of our algorithm's approach while abstracting away implementation details. Additionally, the properties of our algorithm can be established using straightforward inductive arguments. It is also worth noting that adapting traditional sorting algorithms to handle linked list inputs is a non-trivial task, whereas our algorithm can handle such inputs with a minor modification.
For a graph G, the Sombor index and the elliptic Sombor index are defined, respectively, as \[ SO(G) = \sum_{x_i x_j \in E(G)} \sqrt{d_{x_i}<^>2 + d_{x_j}<^>2}, \quad ESO(G) = \sum_{x_i x_j \in E(G)} (d_{x_i} + d_{x_j}) \sqrt{d_{x_i}<^>2 {+} d_{x_j}<^>2}, \] SO(G)=& sum;xixj is an element of E(G)dxi2+dxj2,ESO(G)=& sum;xixj is an element of E(G)(dxi+dxj)dxi2+dxj2, where $ E(G) $ E(G) denotes the edge set of G, and $ d_{x_i} $ dxi is the degree of vertex $ x_i $ xi. The main purpose of this study is to investigate extremal n-vertex c-cyclic graphs for $ c \in \{0,1\} $ c is an element of{0,1}, concerning the Sombor and elliptic Sombor indices, with particular emphasis on the structural parameter of maximum degree Delta. Notably, in each class of c-cyclic graphs with a given Delta, the same graphs attain the maximum value for both the SO and ESO indices. In contrast, the graphs that give the minimum value differ, showing important structural differences between the two indices. These results also remain valid when restricted to molecular graphs.
Decision-making under uncertainty requires flexible models capable of accurately representing expert evaluations and confidence levels. Existing fuzzy soft set-based approaches, including intuitionistic, Pythagorean, and q-rung orthopair fuzzy soft sets, face limitations due to restrictive membership constraints and limited flexibility in handling complex uncertainty. To address these issues, this study proposes a novel Possibility (m,n)-Rung Orthopair Fuzzy Soft Set (P-(m,n)-ROFSS)-based multi-criteria decision-making framework. The proposed approach introduces two independent parameters to control membership and non-membership degrees and incorporates possibility information to better reflect expert confidence. New aggregation operators and score functions are developed, and the practicality of the method is demonstrated through a numerical application for evaluating electronic voting machines. Sensitivity analysis and comparative analysis with existing fuzzy-based methods are conducted to assess robustness and effectiveness. The results indicate that the proposed method provides more flexible uncertainty modelling, stable ranking outcomes, and improved decision reliability compared to traditional approaches, confirming its suitability for solving complex real-world decision-making problems.
An iterative algorithm for constructing an n-node unlabelled binary tree from its inorder-preorder sequence (i-p sequence, tree permutation) is introduced. In the best case, the algorithm performs about 50% fewer comparisons than the best previously known algorithm (Makinen 2000, IJCM). Even in the worst case, it reduces the number of comparisons by about 17%. It requires fewer comparisons for all i-p sequences of length n >= 6. Two theorems on i-p sequences are proven, showing the correctness of the grafting mechanism at the heart of both algorithms, thus closing a gap in the original paper. The use of two virtual nodes as sentinels simplifies boundary conditions when processing i-p sequences. Both algorithms are presented in C. We conclude the new improved algorithm is preferable for constructing binary trees from i-p sequences in O(n) time and space.
In multiprocessor systems, reliable multi-path transmission is crucial for ensuring network quality of service. A set of k spanning trees rooted at the same node r in a network is said to be k node-independent spanning trees (k node-ISTs) if for each node v other than r, the k paths from rto v, with one path in each spanning tree, are internally node-disjoint. This paper studies the problem of constructing k node-ISTs with minimal total path weight, which is NP-complete and has significant application value in fault-tolerant routing, load balancing and secure communication. Considering the limitation of existing algorithms that have been proposed on specific network topologies or for the case k =2 are difficult to extend to the case k >= 3 in general networks, we propose a heuristic algorithm that can construct the required min k node-ISTs in polynomial time O(kn(3)) in any network with n nodes. Experimental simulations indicate that the algorithm can construct high-quality solutions in various random networks and some important specific networks. In addition, this algorithm can effectively meet the practical requirements for fault-tolerance in low-cost and high-connectivity networks.
A sequence $ G_1 $ G1, $ G_2 $ G2, & mldr;, $ G_{k} $ Gk of pairwise edge-disjoint monochromatic subgraphs of a graph G with a red-blue edge colouring is a Ramsey chain in G if $ G_i $ Gi has i edges for $ 1 \le i \le k $ 1 <= i <= k and $ G_i $ Gi is isomorphic to a subgraph of $ G_{i+1} $ Gi+1 for $ 1 \le i \le k-1 $ 1 <= i <= k-1. The subgraphs $ G_i $ Gi are the links of the Ramsey chain and the terminal subgraph $ G_k $ Gk of size k is the target link of the chain. A graph H without isolated vertices is a target graph if there exists a positive integer n such that every red-blue colouring of $ K_n $ Kn results in a Ramsey chain with target link H. For a target graph H, the target Ramsey number $ TR(H) $ TR(H) of H is the minimum positive integer n such that for every red-blue colouring of $ K_n $ Kn, there exists a Ramsey chain having H as its target link. It is shown that $ TR(H) $ TR(H) exists for every graph H without isolated vertices and for the standard (diagonal) Ramsey number $ R(H)=R(H, H) $ R(H)=R(H,H), it follows that $ TR(H)\ge R(H) $ TR(H)>= R(H). The number $ TR(H) $ TR(H) is determined for several graphs H and it is shown that $ TR(H)=R(H) $ TR(H)=R(H) for all these graphs H.
The dependability of multiprocessor systems is essential for facilitating large-scale computing, and the self-diagnosability of the system is the core indicator of its reliability. The cyclic diagnosability of graph G, represented as ct (G), is characterized as the largest size of one faulty vertex set R, provided that after removing R, the remaining graph G - R becomes disconnected and includes no less than two cycles, with each cycle located in a separate connected component. The modified bubble-sort graph is an enhanced variant of the standard bubble-sort graph with a cyclic symmetric structure, which grants its superior topological properties for interconnection networks. In this article, we determine the cyclic diagnosability of the n-dimensional modified bubble-sort graph MBn under the Preparata/Metze/Chien model (PMC-M) and the Maeng/Malek* model (MM*-M) as ct (MBn) = 5n - 10 for n >= 11.
Classical source coding theorems, grounded in Shannon entropy, provide the foundational limits of data compression. However, in various real-world systems - such as those exhibiting long-range dependencies, non-Markovian structures, or multifractal distributions-Shannon entropy may not fully capture the complexity of the source. This paper investigates source coding theorems under generalized entropy measures, particularly Tsallis and & Racute;enyi entropies, and extends classical results to these broader frameworks. We establish necessary and sufficient conditions for source coding under generalized entropy formulations, derive bounds on compression rates, and demonstrate their relevance through examples involving memory-laden and heavy-tailed sources. Furthermore, we explore practical implications in data compression schemes, including entropy coding and universal coding, and validate our theoretical findings via numerical simulations.
For an isolate-free graph G, let $ \gamma (G) $ gamma(G), $ \gamma _t(G) $ gamma t(G), $ \gamma _{pr}(G) $ gamma pr(G), $ \gamma _{t2}(G) $ gamma t2(G) and $ \gamma _{pr2}(G) $ gamma pr2(G) denote domination number, total domination number, paired domination number, semitotal domination number and semipaired domination number, respectively. It is known that $ \gamma (G) \leq \gamma _{t2}(G) \leq \gamma _t(G) \leq \gamma _{pr}(G) $ gamma(G)<=gamma t2(G)<=gamma t(G)<=gamma pr(G) and $ \gamma (G) \leq \gamma _{t2}(G) \leq \gamma _{pr2}(G) \leq \gamma _{pr}(G) $ gamma(G)<=gamma t2(G)<=gamma pr2(G)<=gamma pr(G). Several papers have investigated the complexity of deciding whether a graph attains equality within the above inequalities in general graphs and restricted graph classes such as bipartite graphs and planar graphs. However, the recognition complexity of these equalities in chordal graphs remains open in the literature. In this paper, we address this gap by proving that the decision problem corresponding to each of the above equalities is NP-hard, even when restricted to chordal graphs.
This study presents a numerical treatment of the diffusion model, using the spectral collocation method to discretize the equation in the spatial domain. Temporal discretisation is done through three numerical methods: the implicit forward difference, the Crank-Nicolson and hybrid-block method. We show that the Crank-Nicolson and hybrid block pseudospectral methods are absolutely stable by considering the epsilon-pseudospectra of the resulting matrix from the spatial discretisation. Synthetic noisy signals generated through the imposition of additive white Gaussian noise on some known distributions, such as the Lorentzian peaks and the Voigt profile, are used to evaluate the performance of the discretisation methods. The efficiency of the proposed scheme is quantitatively assessed through the root mean square error (RMSE) and the signal-to-noise ratio (SNR). These metrics evaluate the extent to which the numerical discretisation techniques successfully remove distortions from the signals. Results confirm the diffusion model and numerical methods denoise distorted signals.
The g-extra connectivity of a connected graph G, denoted by kappa(g)(G), is a generalization of classical connectivity which related to the integer g. In this paper, we first construct a novel network named semicrossing cube SCQ(n). We prove that kappa 4(SCQ(n)) = 5n - 13. This gives an answer to a problem proposed by Zhou (J. Comput. System Sci., 88(2017) 208-219).
Let $s,m$ be the positive integers and $p$ be any prime number. Next, let $GR(p^s,p^{sm})$ be a Galois ring of characteristic $p^s$ and cardinality $p^{sm}$. In the present paper, we explore the construction of Cauchy MDS matrices over Galois rings. Moreover, we introduce a new approach that considers nilpotent elements and Teichmüller set of Galois ring $GR(p^s,p^{sm})$ to reduce the number of entries in these matrices. Furthermore, we construct $p^{(s-1)m}(p^m-1)$ distinct functions with the help of Frobenius automorphisms. These functions preserve MDS property of matrices. Finally, we prove some results using automorphisms and isomorphisms of the Galois rings that can be used to generate new Cauchy MDS matrices.
In 2019, V. A. Roman'kov introduced the concept of marginal sets for groups. He developed a theory of marginal sets and demonstrated how these sets can be applied to improve some key exchange schemes. In this paper, we extend his ideas and introduce the concept of marginal sets for semigroups and semirings. For tropical matrix semigroups and semirings, we describe how some marginal sets can be constructed. We apply marginal sets to improve some key exchange schemes over semigroups.
Seeking and establishing sufficient conditions to ensure Hamiltonicity of connected graphs is crucial and valuable because of the classical NP-complete attribute. Quasi-Laplacian energy, a graph invariant in terms of the quasi-Laplacian spectrum, is a powerful tool in the resolution process of Hamilton-related problems. Let G be an n-vertex connected graph with quasi-Laplacian eigenvalues mu(1) >= E-Q(G) = Sigma(n) (i=1) mu(2) >= ... >= mu(n) >= 0. The quasi-Laplacian energy of G is defined as Sigma(n)(i=1) mu(i)(2) . In this paper, we suggest some sufficient conditions in terms of EQ(G) for graphs to be k-hamiltonian, Hamiltonian, kleaf-connected, Hamilton-connected and k-connected, respectively.
