The present paper gives a detailed study of the structural theory of triple \(\theta\)-skew cyclic codes where the codes are over \(\mathbb{F}_q\). We give a complete characterization of these codes, focusing on their representation as modules. We identify the generator polynomials for the triple skew-cyclic codes as well as those of their duals. We explore the properties of these generator polynomials and their relationship to the code’s structure. Additionally,To illustrate our approach, we give concrete instances of triple \(\theta\)-skew cyclic codes to demonstrate how these structures can behave in practice. The special instances we present reveal that such codes are capable of achieving strong parameters under certain conditions.

Let \(P_{k+1}\) denote a path of length \(k\), let \(S_{m}\) denote a star with \(m\) edges, and let \(K_{n}(\lambda)\) denote the complete multigraph on \(n\) vertices in which every edge is taken \(\lambda\) times. In this paper, we prove that the necessary conditions are also sufficient for a \(\{P_{4}, S_{4}\}\)-decomposition of \(K_{n}(\lambda)\).

A \(\{P_5^{\alpha}, S_5^{\beta}, Y_5^{\gamma}\}\)-decomposition of a graph is a partition of its edge set into \(\alpha\) copies of \(P_5\), \(\beta\) copies of \(S_5\), and \(\gamma\) copies of \(Y_5\), where \(P_5\), \(S_5\), and \(Y_5\) denote the three non-isomorphic trees of order five. In this paper, we study the existence of a \(\{P_5^{\alpha}, S_5^{\beta}, Y_5^{\gamma}\}\)-decomposition of the complete bipartite graph \(K_{m,n}\) for \(m\geq 4\) and \(n\geq 2\), and of the complete graph \(K_n\) for \(n\geq 8\). In fact, we establish necessary and sufficient conditions for the existence of such decompositions in \(K_{m,n}\) and \(K_n\).

Given a connected graph \(G\) and a configuration \(D\) of pebbles on \(V(G)\), a pebble move consists of removing two pebbles from one vertex and placing one pebble on an adjacent vertex. A monophonic path is a longest chordless path between two non-adjacent vertices \(u\) and \(v\). The line segment that connects two vertices on a curve is known as a chord. The monophonic distance between \(u\) and \(v\) is the number of vertices in the longest \(u\)–\(v\) monophonic path, denoted by \(d_{\mu}(u,v)\) in \(G\). The monophonic pebbling number (MPN) of \(G\) is the least number of pebbles needed to guarantee that, from any distribution of pebbles on a graph \(G\), one pebble can be moved to any specified vertex using monophonic paths through pebbling moves. The monophonic \(t\)-pebbling number (MtPN) of \(G\) is the least number of pebbles needed to guarantee that, from any distribution of pebbles, \(t\) pebbles can be moved to any specified vertex using monophonic paths. In this article, we determine the \(MPN\) and \(MtPN\) of Dutch windmill graphs, square of cycles, tadpole graphs, lollipop graphs, double star path graphs, and fuse graphs, and we also discuss their \(t\)-pebbling versions.

In this paper, we consider circulant graphs obtained from the complete graph \(K_N\) by deleting all edges belonging to a prescribed distance class. We study, in a unified manner, the effective resistance, the expected hitting time, the number of spanning trees, and the number of two-component spanning forests of these graphs. For general distance-class deletions, these quantities admit natural spectral representations in terms of the Laplacian eigenvalues. However, such representations typically remain at the level of finite Fourier sums, and concise closed forms are not expected in general. We focus on the case of a single deleted distance class. When the number of vertices \(N\) is odd and \(\gcd(r,N)=1\), the graph \(G_{N,r}\) is isomorphic to \(G_{N,1}\). In this setting, we derive explicit exponential-type formulas for the effective resistance and the number of spanning trees, and obtain corresponding closed expressions for two-component spanning forests and expected hitting times. Our results show that the case \(r=2\) is not essentially new, but follows from a general isomorphism structure underlying distance-class deletions. We also clarify the relation of our formulas to earlier results on the complete graph with a Hamiltonian cycle removed, and provide a unified derivation within a spectral framework. Moreover, by asymptotic analysis, we show that the ratio \(\tau(G_{N,1})/\tau(K_N)\) converges to \(e^{-2}\) as \(N \to \infty\).

A graph \(G\) with vertex set \(V(G)\) and edge set \(E(G)\) is said to have an odd prime labeling if there exists a bijection \(f:V(G)\to \{1,3,5,\dots,2n-1\}\), where \(n=|V(G)|\), such that \(\gcd(f(x),f(y))=1\) for every edge \(xy\in E(G)\). In this paper, we study odd prime labelings of graphs arising from duplication operations on graph elements. We obtain several results for graphs derived from the path graph \(P_n\), the cycle graph \(C_n\), and the star graph \(K_{1,n}\) under various vertex- and edge-duplication constructions.

What are the collections of sets \({A}_i\subset\mathbb{Z}\) such that any \(n\in\mathbb{Z}\) has exactly one representation as \(n=a_0+a_1+\dotsb\) with \(a_i\in{A}_i\)? The answer for \(\mathbb{N}_0\) instead of \(\mathbb{Z}\) is given by a theorem of de Bruijn. We describe a family of natural candidate collections for \(\mathbb{Z}\), which we call canonical collections. Translating the problem into the language of dynamical systems, we show that the question of whether the sumset of a canonical collection covers the entire \(\mathbb{Z}\) is difficult: specifically, there is a collection for which this question is equivalent to the Collatz conjecture, and there is a well-behaved family of collections for which this question is equivalent to the universal halting problem for Fractran and is therefore undecidable.

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.