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.
We study the Equivalent Local Sequence Problem (ELSP), which consists in recovering an explicit sequence of local complementations that transforms a graph into a locally equivalent one. Focusing on directed Paley graphs, we establish that local complementations commute and induce a free action of an elementary abelian 2-group. The stabilizer condition is reformulated as a system of convolution equations and analyzed through Fourier techniques over finite fields, leading to a proof of stabilizer triviality. As a consequence, each graph in the orbit admits a unique subset encoding, and ELSP reduces to solving a linear inversion problem over 𝔽2. This characterization completely resolves ELSP for directed Paley graphs, provides a polynomial-time inversion algorithm and highlights structural features that may support future developments in cryptographic frameworks and quantum graph-state models.
Given a configuration of pebbles on the edges of a connected graph G, an edge pebbling move is defined as the removal of two pebbles off an edge and placing one on an adjacent edge. The domination cover edge pebbling number of a graph G is the minimum number of pebbles required such that the set of edges that contain pebbles form an edge dominating set S of G, for the initial configuration of pebbles can be altered by a sequence of pebbling moves and it is denoted by ψe(G) for a graph G. In this paper, we determine ψe(G) for Generalized Petersen graph, Jewel graph and Triangular snake graph.
In this paper we study group divisible designs (GDDs) with block size 4 and two groups of different sizes when λ2 = 1. We obtain necessary conditions for the existence of such GDDs and prove that these necessary conditions are sufficient in several cases. Further, we present general constructions using resolvable designs.
In 1940, Birkhoff posed an open problem of counting all finite lattices on n elements. Recently, Bhavale counted all non-isomorphic lattices on n elements, containing up to four reducible elements, and having nullity up to three. Further, Aware and Bhavale counted all non-isomorphic lattices on n elements, containing up to five comparable reducible elements, and having nullity up to three. In this paper, we count all non-isomorphic lattices on n elements, containing five reducible elements, and having nullity three.
The unit graph of a commutative ring with a non-zero identity is a graph with vertices as ring elements, and there is an edge between two distinct vertices if their sum is a unit. This study investigates the decomposition of the unit graph by examining its induced subgraphs and analyze key graph invariants, such as connectivity, diameter, and girth, for a finite local ring. We further decompose the unit graph of certain finite commutative rings into fundamental structures, such as cycle and star graphs.
A mapping of the set of undirected simple (loopless) graphs to itself is a linear operator if it maps the edgeless graph to the edgeless graph and maps the union of graphs to the union of their images. A linear operator preserves a set if it maps that set to itself. We study linear operators that map sets defined by the restriction of their chromatic number. For example the set of all graphs whose chromatic number is at least \(k\) for some fixed \(3\leq k\leq n\). We show these linear operators must be vertex permutations.
The first Zagreb index of a graph \(G\) is defined as \(\sum\limits_{u \in V} d_G^2(u)\), where \(d_G(u)\) is the degree of vertex \(u\) in \(G\). The algebraic connectivity of a graph \(G\) is defined as the second smallest eigenvalue of the Laplacian matrix of \(G\). Using Wagner’s inequality, we in this paper first obtain an upper bound for the algebraic connectivity that involves the first Zagreb index of a graph. Following the ideas of obtaining the upper bound, we present sufficient conditions involving the first Zagreb index and the algebraic connectivity for some Hamiltonian properties of graphs.
For a graph \(G\), let \(la(G)\) denote the linear arboricity of \(G\) and \(\Delta(G)\) denote the maximum degree of \(G\). The famous linear arboricity conjecture was made by Akiyama, Exoo, and Harary [Covering and packing in graphs. IV. Linear arboricity] in 1981. It asserts that \(la(G) \leq \Bigl\lceil\frac{\Delta(G)+1}{2}\Bigr\rceil\). In this paper, we prove the linear arboricity conjecture for products of a path and a complete graph, and for products of a path and a tree.
Let \(G\) be a connected graph. The center function defined on \(G\) yields a set of vertices that minimizes the maximum distance from the given input vertices. Through axiomatic characterization of the center function, we identify the specific axioms that characterize its behavior on connected graphs. Universal axioms encompass the properties satisfied by the center function on all connected graphs. However, for some graphs, the center function cannot be fully characterized using universal axioms alone. To address this, a set of graph class-specific axioms, known as non-universal axioms, was introduced. In the case of book graphs (Cartesian product of star graph \(K_{1,n}\) and path \(P_2\)), the center function cannot be adequately characterized using known universal axioms. Therefore, in this context, we find an axiomatic characterization of the center function on book graphs using the universal axioms and one newly introduced Cycle Consensus \((CC)\) axiom.
We study a search problem on capturing a moving target on an infinite real line. Two autonomous mobile robots (which can move with a maximum speed of 1) are initially placed at the origin, while an oblivious moving target is initially placed at distance d from the origin. The robots can move along the line in either direction, but the target is oblivious, cannot change direction, and moves either away from or toward the origin at a constant speed v. Our aim is to design efficient algorithms for two robots to capture the target. The target is captured only when both robots are co-located with it. The robots communicate only face-to-face (F2F), meaning they can exchange information only when co-located. We design algorithms under various knowledge scenarios regarding d, v, and the target’s direction of movement. We analyze competitive ratios, i.e., the capture time versus the optimal full-knowledge scenario, and show that our strategies use at most three direction changes.
We present a theorem which characterizes the class of line graphs of directed graphs. The characterization is an analogue of both the characterization of line graphs by Krausz (1943) and of directed line graphs of directed graphs by Harary and Norman (1960). Our characterization simplifies greatly in the case that the graph is bipartite. This and another result which we present draws attention to the special case of bipartite line graphs of directed graphs. As a result we explore the problem of finding the complete list of forbidden subgraphs for the class of bipartite line graphs of directed graphs. It appears, however, that this problem is extremely difficult. We do find two infinite families of forbidden subgraphs as well as several other illustrative examples.
A block graph is a graph in which each block (maximal biconnected subgraph) is a clique. A block graph can, equivalently, be seen as a tree of cliques where the blocks form complete subgraphs connected to each other in a tree-like, hierarchical structure. Such graphs provide a good conceptual and computational framework in designing, analyzing, and optimizing resilient and efficient power networks. In view of the above applications, this paper investigates the \(k\)-fault tolerant power domination number of block graphs. Also, we obtained a lower bound for \(k\)-fault tolerant power domination number when an edge in the graph is contracted.
