In this paper we study group divisible designs (GDDs) with block size 4 and two groups of different sizes when \(\lambda_{2}=1\). 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.

The Narumi-Katayama index of a graph is defined as the product of the degrees of all vertices in the graph. In this paper, we obtain upper bounds of the Narumi-Katayama index of a connected graph. We further present sufficient conditions based on the Narumi-Katayama index and other graph invariants for Hamiltonian and traceable graphs.

Christoph, Müyesser and Wigderson recently asked whether an approximate form of the Erdős–Sós conjecture is robust under random edge deletions. We establish a density-sensitive transference theorem that converts global resilience for bounded-degree trees in sparse random graphs into resilient forest universality in arbitrary dense host graphs. More precisely, if \(F\) is an \(N\)-vertex graph of edge density \(\lambda\) bounded away from zero and \(p\in[K/N,1]\), then, with probability \(1-o(1)\) uniformly over the host and the percolation parameter, every subgraph obtained from \(F_p\) by deleting at most an \(\alpha\)-fraction of its edges contains every bounded-degree forest on at most \(((1-\alpha)\lambda-\xi)N\) vertices. Consequently, for every \(c>0\), \(L\ge1\), fixed \(D\) and \(\alpha<c\), every graph \(F\) on at most \(Ld\) vertices with average degree at least \(d\) has the property that, after percolation at any rate \(p\in[K/d,1]\) and any subsequent deletion of at most an \(\alpha\)-fraction of the surviving edges, the remaining graph is universal for all forests on at most \((1-c)d\) vertices and maximum degree at most \(D\). This includes vertex-disjoint packings of any prescribed collection of bounded-degree trees of that total order. We further obtain forest-universality results for graphs whose connected components have vertex-cover number at most \(Cd\), and for graphs that can be brought into this form by deleting sufficiently few edges on the \(dn\)-scale.

This paper investigates the combinatorial properties, invariants, and structures arising from the action of the direct product \(S_n \times A_n\) on the Cartesian product \(X^{(2)} \times Y^{(2)}\), where \(X^{(2)}\) and \(Y^{(2)}\) denote the sets of unordered 2-element subsets of two distinct sets \(X\) and \(Y\), each of cardinality \(n\). Using the Orbit-Stabilizer Theorem, we establish that the action is transitive for all \(n \ge 2\) and, through block theory, demonstrate that it is imprimitive for all \(n \ge 3\). By determining the orbits of the stabilizer of a fixed element, we compute the rank of the action to be \(9\) and explicitly enumerate the eight non-trivial subdegrees for \(n \ge 5\) as: \(2(n-2)\), \(2(n-2)\), \(\frac{(n-2)(n-3)}{2}\), \(\frac{(n-2)(n-3)}{2}\), \(4(n-2)^2\), \((n-2)^2(n-3)\), \((n-2)^2(n-3)\), and \(\frac{(n-2)^2(n-3)^2}{4}\). A combinatorial proof establishes that all suborbits are self-paired for \(n\ge 5\). We construct the eight non-trivial suborbital graphs corresponding to these suborbits and analyze their fundamental graph-theoretic properties, including connectedness, regularity, vertex degrees, and girth. The results extend the classical theory of permutation groups acting on combinatorial objects and provide a foundation for potential applications in coding theory, cryptography, and control systems.

For each integer \(m\geq -1\), we study the family of matrices \(\mathsf{S}^{(m)}(n)=\bigl[S(i+j+m, j)\bigr]_{1\leq i, j\leq n},\) whose entries are Stirling numbers of the second kind. We derive several explicit matrix decompositions of \(\mathsf{S}^{(m)}(n)\) in terms of the classical Stirling matrix and certain explicitly constructed upper triangular matrices. As a consequence, we obtain the closed-form determinant formula \(\det \mathsf{S}^{(m)}(n)=\prod_{i=1}^{n} i^{\,i+m},\) which extends several previously known determinant evaluations. We also establish several identities involving both the Stirling numbers of the first and second kinds.