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 ψ_e(G) for a graph G. In this paper, we determine ψ_e(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.