We extend the study of link-irregular graphs to directed graphs (digraphs), where a digraph is link-irregular if no two vertices have isomorphic directed links. We establish that link-irregular digraphs exist on n vertices if and only if n ≥ 5, and prove that their underlying graphs must contain 3-cycles. We conjecture that link-irregular tournaments exist if and only if n ≥ 6, providing explicit constructions for n ≤ 8 and computational verification for n ≤ 100. We derive lower bounds on the minimum degree and outdegree required for link-irregularity, establish that almost all link-irregular digraphs are nonplanar, and prove that any link-irregular orientable graph admits a link-irregular labeling. Additionally, we construct explicit examples of link-irregular digraphs with constant outdegree and regular tournaments.

In food processing, effective optimization of process parameters can improve product quality, reduce production costs, and shorten production cycles. This paper improves the traditional particle swarm optimization algorithm by introducing an adaptive learning factor and an elite particle variation strategy, thereby balancing global search and local convergence while preventing particle aggregation and local optima. Using kimchi as a case study, an optimization model including fermentation temperature and other variables is developed, and the improved algorithm is applied to the multi-objective optimization of processing parameters to achieve optimal product quality. Experimental results are compared with those of the traditional particle swarm algorithm and the response surface method. The findings show that the improved model requires only 43 iterations and reduces final risk loss by 14.26%, outperforming the other two methods, which require 52 and 100 iterations, respectively. Thus, the improved particle swarm optimization model can effectively shorten the optimization cycle and reduce the cost of food-processing parameter optimization.

This work introduces two algebraic variants of the Massey-Omura cryptosystem based on newly defined generalized (k,t)-Jacobsthal p-numbers and their extensions to finite groups. We first generalize the classical Jacobsthal recurrence and establish structural properties including periodicity, invertibility conditions, and recurrence behavior modulo finite integers. These results are then extended to group-theoretic settings, where we construct the corresponding (k,t)-Jacobsthal sequences in specific finite groups and derive their sequence periods. Leveraging these algebraic foundations, we propose two Massey-Omura-type encryption schemes in which private exponents are selected from the generalized Jacobsthal sequences. We formally prove the correctness of both constructions and analyze the implications of periodicity on exponent invertibility and protocol feasibility. The proposed schemes do not introduce new hardness assumptions beyond those inherent in the underlying platform group. Instead, they provide a mathematically structured alternative to classical exponent selection in three-pass protocols. The results highlight a new connection between recurrence-defined sequences and multiplicative exponentiation in finite groups, offering an algebraically motivated direction for exploring generalized exponent families in symmetric and non-abelian cryptosystems.

Given any integers \(q\ge 2\) and \(p\ge 3\), a multidimensional array \[[m(i_1,i_2, \dots, i_q)\colon 1\le i_1\le p, \dots , 1\le i_q\le p],\] with non-repeated entries from the set \(\{1,2,\dots, p^q\}\) will be called a \(q\) dimensional magic hyper-square of order \(p\) if the sum of the numbers in any of its axes or diagonals is \(p(p^q+1)/2\). In this paper, we study odd-order uniform step magic hyper-squares of the form \[m(i_1, \dots, i_q)=1+\sum\limits_{j=1}^{q} p^{j-1}\left[\left(\sum\limits_{k=1}^qa_{j,k} i_k+d_j\right) \bmod p \right] .\] We derive necessary and sufficient conditions for these coefficients to generate magic hyper-squares and use a specific choice of coefficients to get new formula that generates magic hyper-squares of all odd orders greater than two.

In this paper, we consider two ways of breaking a graph’s symmetry: distinguishing labelings and fixing sets. A distinguishing labeling ϕ of G colors the vertices of G so that the only automorphism of the labeled graph (G, ϕ) is the identity map. The distinguishing number of G, D(G), is the fewest number of colors needed to create a distinguishing labeling of G. A subset S of vertices is a fixing set of G if the only automorphism of G that fixes every element in S is the identity map. The fixing number of G, Fix(G), is the size of a smallest fixing set. A fixing set S of G can be translated into a distinguishing labeling ϕ_S by assigning distinct colors to the vertices in S and assigning another color (e.g., the “null” color) to the vertices not in S. Color refinement is a well-known efficient heuristic for graph isomorphism. A graph G is amenable if, for any graph H, color refinement correctly determines whether G and H are isomorphic or not. Using the characterization of amenable graphs by Arvind et al. as a starting point, we show that both D(G) and Fix(G) can be computed in O((|V(G)|+|E(G)|)log |V(G)|) time when G is an amenable graph.