Let \(p > 5\) be a prime positive integer, \(m\) and \(s\) be positive integers. We classify the negacyclic codes of length \(5p^s\) over \(R= \mathbb{F}_{p^m}+u\mathbb{F}_{p^m}\), with \(u^2=0\) using the factorisation of cyclotomic polynomials, and we investigate their Hamming distances.

Dominator coloring is a fascinating type of proper coloring where vertices are assigned colors so that every vertex in the graph is within the closed neighborhood of at least one vertex from each color class. The smallest number of colors needed for a dominator coloring is called the dominator chromatic number. In this paper, a new graph product called the closed extended neighborhood corona of two graphs is introduced and its dominator chromatic number for any pair of connected graphs is determined. Also, the dominator chromatic numbers for the extended corona of a path with any graph and a cycle with any graph are derived. Additionally, the dominator chromatic number for the closed neighborhood corona of any two graphs is established.

The article investigates the domination polynomial of generalized friendship graphs. The domination polynomial captures the number of dominating sets of each cardinality in a graph and is known to be NP-complete to compute for general graphs. We establish the log-concave and unimodal properties of these polynomials, and determine their peaks. Furthermore, we analyze the distribution of the zeros of aforesaid polynomial and identify their region in the complex plane. Several open problems are proposed for future exploration.

The generalized Petersen graph \(G(n,k)\) is a cubic graph with vertex set \(V(G(n,k))=\{v_i\}_{0 \leq i < n} \cup \{w_i\}_{0 \leq i < n}\) and edge set \(E(G(n,k))=\{v_i v_{i+1}\}_{0 \leq i < n} \cup \{w_i w_{i+k}\}_{0 \leq i < n} \cup \{v_i w_i\}_{0 \leq i < n}\) where the indices are taken modulo \(n\). Schwenk found the number of Hamiltonian cycles in \(G(n,2)\), and in this article we present initial conditions and linear recurrence relations for the number of Hamiltonian cycles in \(G(n,3)\) and \(G(n,4)\). This is attained by introducing \(G'(n,k)\), which is a modified version of \(G(n,k)\), and a subset of its subgraphs which we call admissible, and which are partitioned into different classes in such a manner that we can find relations between the number of admissible subgraphs of each class. The classes and their relations define a directed graph such that each strongly connected component is of a manageable size for \(k=3\) and \(k=4\), which allows us to find linear recurrence relations for the number of admissible subgraphs in each class in these cases. The number of Hamiltonian cycles in \(G(n,k)\) is a sum of the number of admissible subgraphs of \(G'(n,k)\) over a certain subset of the classes.

Perfect codes in the \(n\)-dimensional grid \(\Lambda_n\) of the lattice \(\mathbb{Z}^n\) (\(0<n\in\mathbb{Z}\)) and its quotient toroidal grids were obtained via the truncated distance in \(\mathbb{Z}^n\) given between \(u=(u_1,\cdots,u_n)\) and \(v=(v_1, \ldots,v_n)\) as the graph distance \(h(u,v)\) in \(\Lambda_n\), if \(|u_i-v_i|\le 1\), for all \(i\in\{1, \ldots,n\}\), and as \(n+1\), otherwise. Such codes are extended to superlattice graphs \(\Gamma_n\) obtained by glueing ternary \(n\)-cubes along their codimension 1 ternary subcubes in such a way that each binary \(n\)-subcube is contained in a unique maximal lattice of \(\Gamma_n\). The existence of an infinite number of isolated perfect truncated-metric codes of radius 2 in \(\Gamma_n\) for \(n=2\) is ascertained, leading to conjecture such existence for \(n>2\) with radius \(n\).