In this paper, we investigate the metric dimension of generalized Petersen graphs \(P(n,3)\), providing a partial answer to an open problem posed in [8]: whether \(P(n,m)\) for \(n \geq 7\) and \(3 \leq m \leq \left\lfloor \frac{n-1}{2} \right\rfloor\) constitutes a family of graphs with constant metric dimension. Specifically, we prove that the metric dimension of \(P(n,3)\) equals \(3\) for \(n \equiv 1 \pmod{6}\), \(n \geq 25\), and equals \(4\) for \(n \equiv 0 \pmod{6}\), \(n \geq 24\). For remaining cases, four judiciously chosen vertices suffice to resolve all vertices of \(P(n,3)\), implying \(\dim(P(n,3)) \leq 4\), except when \(n \equiv 2 \pmod{6}\), in which case \(\dim(P(n,3)) \leq 5\).

Using subspaces of the finite field \(GF(q^{2^k})\) over \(GF(q)\), we construct new classes of external difference families.

Let \(M = \{v_1, v_2, \ldots, v_n\}\) be an ordered set of vertices in a graph \(G\). Then, \((d(u, v_1), d(u, v_2), \ldots, d(u, v_n))\) is called the \(M\)-coordinates of a vertex \(u\) of \(G\). The set \(M\) is called a \({metric\; basis}\) if the vertices of \(G\) have distinct \(M\)-coordinates. A minimum metric basis is a set \(M\) with minimum cardinality. The cardinality of a minimum metric basis of \(G\) is called the minimum metric dimension. This concept has wide applications in motion planning and robotics. In this paper, we solve the minimum metric dimension problem for Illiac networks.

For a graph \(G\) and a non-zero real number \(\alpha\), the graph invariant \(S_\alpha(G)\) is the sum of the \(\alpha^th\) power of the non-zero signless Laplacian eigenvalues of \(G\). In this paper, we obtain sharp bounds of \(S_\alpha(G)\) for a connected bipartite graph \(G\) on \(n\) vertices and a connected graph \(G\) on \(n\) vertices having a connectivity less than or equal to \(k\), respectively, and propose some open problems for future research.

In this paper we determine the scores of locally transitive tournaments and conversely, for such score we construct all locally transitive tournments having this score. This allows us to establish, for a given matrix, a test for the locally transitive property.

A graph is called a cover graph if it is the underlying graph of the Hasse diagram of a finite partially ordered set. The direct product \(G \times H\) of graphs \(G\) and \(H\) has vertex set \(V(G) \times V(H)\) and edge set \(E(G \times H) = \{ (g_i, h_s)(g_j, h_t) \mid g_ig_j \in E(G) \text{ and } h_sh_t \in E(H) \}\). We prove that the direct product \(M_m(G) \times M_n(H)\) of the generalized Mycielskians of \(G\) and \(H\) is a cover graph if and only if \(G\) or \(H\) is bipartite.

For a primitive digraph \(D\) of order \(n\) and a positive integer \(m\) such that \(1 \leq m \leq n\), we define the \(m\)-competition index of \(D\), denoted by \(k_m(D)\), as the smallest positive integer \(k\) such that distinct vertices \(v_1, v_2, \ldots, v_m\) exist for each pair of vertices \(x\) and \(y\) with \(x \rightarrow^k v_i\) and \(y \rightarrow^k v_i\) for \(1 \leq i \leq m\) in \(D\). In this paper, we investigate the \(m\)-competition index of regular or almost regular tournaments.

A digraph \(D\) with \(e\) edges is labeled by assigning a distinct integer value \(\theta(v)\) from \(\{0, 1, \ldots, e\}\) to each vertex \(v\). The vertex values, in turn, induce a value \(\theta(u,v) = \theta(v) – \theta(u) \mod (e + 1)\) on each edge \((u,v)\). If the edge values are all distinct and nonzero, then the labeling is called a \emph{graceful labeling} of a digraph. Bloom and Hsu conjectured in 1985 that “all unicyclic wheels are graceful.” In this paper, we prove this conjecture.

Let \(m \geq 2\) be an integer and let \(G\) be a finite Abelian group of order \(p^n\), where \(p\) is an odd prime and \(n\) is a positive integer. In this paper, we derive necessary and sufficient conditions for the existence of an \(m\)-adic splitting of \(G\), and hence for the existence of polyadic codes (as ideals in an Abelian group algebra) of length \(p^n\). Additionally, we provide an algorithm to construct all \(m\)-adic splittings of \(G\). This work generalizes the results of Ling and Xing \([9]\) and Sharma, Bakshi, and Raka \([14]\).

Let \(G\) be a finite abelian group. The critical number \(cr(G)\) of \(G\) is the least positive integer \(m\) such that every subset \(A \subseteq G \setminus \{0\}\) of cardinality at least \(m\) spans \(G\), i.e., every element of \(G\) can be expressed as a nonempty sum of distinct elements of \(A\). Although the exact values of \(cr(G)\) have been recently determined for all finite abelian groups, the structure of subsets of cardinality \(cr(G) – 1\) that fail to span \(G\) remains characterized except when \(|G|\) is even or \(|G| = pq\) with \(p, q\) primes. In this paper, we characterize these extremal subsets for \(|G| \geq 36\) and \(|G|\) even, or \(|G| = pq\) with \(p, q\) primes and \(q \geq 2p + 3\).