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.

This paper studies the classification problem of block-transitive \( t \)-designs. Let \(\cal D = (\mathcal{P}, \mathcal{B}) \) be a non-trival \( t\)-\((v,k,\lambda) \) design with \( \lambda \leq 5 \), and let \( G \) be a block-transitive, point-primitive automorphism group of \(\cal D \). We prove that if \( \text{Soc}(G) \) is a sporadic simple group, then up to isomorphism, there are exactly 15 such designs \( \cal D \).

The Mostar invariants are newly introduced bond-additive, distance-related descriptors that compute the degree of peripherality of specific edges as well as the entire graph. These invariants have attracted significant attention in both classical applications of chemical graph theory and studies of complex networks. They have proven to be useful for exploring the topological aspects of these networks. For a graph ℋ, the edge Mostar index Mo_e is defined as the sum of the magnitudes of the differences between m_ℋ(x) and m_ℋ(g) across all edges xg of ℋ. Here, m_ℋ(g) (or m_ℋ(x)) represents the cardinality of the edges in ℋ that are closer to g (or x) than x (or g). In this paper, we determine the trees that maximize and minimize the edge Mostar index for fixed order, diameter, and number of pendent vertices. Sharp upper and lower bounds for this index are established, and the corresponding extremal trees are characterized. Moreover, the correlation of the edge Mostar index with certain physicochemical properties is examined.

The design whose blocks consist of all \(k\)-element multisets drawn from a \(v\)-set, denoted \(M(v,k)\), is a classical example of a balanced \((k+1)\)-ary design. Although its parameters are well known, existing derivations often rely on general multiset design theory. This paper gives unified elementary derivations of the parameters \(b\), \(r\), and \(\lambda\) using stars-and-bars and double counting. We exhibit a natural multiplicity-layer decomposition: removing \(s\) copies of a fixed point from all blocks in which it has multiplicity exactly \(s\) yields a family of subdesigns naturally in bijection with \(M(v-1,k-s)\). This viewpoint clarifies the recursive structure underlying complete multiset designs. Finally, the multiplicity vectors of blocks of \(M(v,k)\) form a \((k+1)\)-ary code of length \(v\) with constant coordinate sum \(k\) and minimum Hamming distance \(2\), achieving size \(\binom{v+k-1}{k}\).

The Explorer-Director game, first introduced by Nedev and Muthukrishnan (2008), simulates a Mobile Agent exploring a ring network with an inconsistent global sense of direction. Two players, the Explorer and the Director, jointly control a token’s movement on the vertices of a graph G with initial location v. Each turn, the Explorer calls any valid distance, d, aiming to maximize the number of vertices the token visits, and the Director moves the token to any vertex distance d away aiming to minimize the number of visited vertices. The game ends when no new vertices can be visited, assuming optimal play, and we denote the total number of visited vertices by f_d(G, v). Here we study a variant where, if the token is on vertex u, the Explorer is allowed to select any valid path length, ℓ, and the Director now moves the token to any vertex v such that G contains a uv path of length ℓ. The corresponding parameter is f_p(G, v). In this paper, we explore how far apart f_d(G, v) and f_p(G, v) can be, proving that for any n there are graphs G and H with f_p(G, v) − f_d(G, v) > n and f_d(H, v) − f_p(H, v) > n.

We study edge partitions of a bipartite graph into induced-2K₂-free bipartite graphs, i.e. into Ferrers (chain) graphs. We define fp(G) as the minimum number of parts in such a partition. We prove general lower and upper bounds in terms of induced matchings and Dilworth widths of neighborhood posets. We compute the parameter exactly for paths and even cycles, and we exhibit separations showing that the induced-matching lower bound and the width upper bound can both be far from tight. We also record a simple host-induced conflict-graph lower bound, present a 0–1 matrix viewpoint, and add some complexity remarks.

We present a proof of a conjecture of Goh and Wildberger on the factorization of the spread polynomials. We indicate how the factors can be effectively calculated and exhibit a connection to the factorization of Fibonacci numbers into primitive parts.

An edge-coloring of a graph \(G\) with natural numbers \(1,2,\ldots\) is called a sum edge-coloring if the colors of edges incident to any vertex of \(G\) are distinct and the sum of the colors of the edges of \(G\) is minimum. The edge-chromatic sum of a graph \(G\) is the sum of the colors of edges in a sum edge-coloring of \(G\). In general, the problem of finding the edge-chromatic sum of an \(r\)-regular (\(r\geq 3\)) graph is \(NP\)-complete. In this paper we provide some bounds on the edge-chromatic sums of various products of graphs. In particular, we give tight upper bounds on the edge-chromatic sums of tensor, strong tensor, Cartesian, strong products and composition of graphs. We also determine the edge-chromatic sums and edge-strengths of the Cartesian products of regular graphs and paths (cycles) with an even number of vertices. Finally, we determine the edge-chromatic sums and edge-strengths of grids, cylinders, and tori.

Generalised nice sets are defined as subsets of edges of the extended Fano plane satisfying a certain absorbing property. The classification up to collineations, purely combinatorial in nature, provides 245 generalised nice sets. In turn, this gives rise to new Lie algebras obtained by modifying the bracket of homogeneous elements on an initial \(\mathbb Z_2^3\)-graded Lie algebra.