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 \(n\).

A graph \(G=(V,E)\) is said to be a \(k\)-threshold graph with thresholds \(\theta_1<\theta_2<…<\theta_k\) if there is a map \(r: V \longrightarrow \mathbb{R}\) such that \(uv\in E\) if and only if the number of \(i\in[k]\) with \(\theta_i\le r(u)+r(v)\) is odd. The threshold number of \(G\), denoted by \(\Theta(G)\), is the smallest positive integer \(k\) such that \(G\) is a \(k\)-threshold graph. In this paper, we determine the exact threshold numbers of cycles by proving \[\Theta(C_n)=\begin{cases} 1 & if\ n=3, \\ 2 & if\ n=4, \\ 4 & if\ n\ge 5, \end{cases}\] where \(C_n\) is the cycle with \(n\) vertices.

Let G = (V, E) be a simple connected graph and W ⊆ V. For v ∈ V, the representation multiset or m-code of v is the multiset r_m(v) = {d(v, w) ∣ w ∈ W}. If no two vertices in G have equal m-codes, then W is called an m-resolving set of G. The multiset dimension md(G) of G is the minimum possible cardinality of an m-resolving set of G, if such a set exists. If G does not possess an m-resolving set, then we say that G has infinite multiset dimension. In this paper, we show that all cylindrical graphs P_m ▫ C_n, where m, n ≥ 3, have finite multiset dimension. In particular, we show that md(P_m ▫ C_n) ≤ 4 if m ≥ 6 and n ≥ 3, or if m ≥ 3 and n ≥ 12. Moreover, if m ≥ 3 and n ≥ 8m + 1, we show that P_m ▫ C_n has multiset dimension 3.

In 2020 Bhavale and Waphare introduced the concept of a nullity of a poset as nullity of its cover graph. According to Bhavale and Waphare, if a dismantlable lattice of nullity k contains r reducible  elements then 2 ≤ r ≤ 2k. In 2003 Pawar and Waphare counted all non-isomorphic lattices on n elements having nullity one, containing exactly two reducible elements. Recently, Bhavale and Aware counted all non-isomorphic lattices on n elements having nullity two, containing up to three  reducible elements. In this paper, we count up to isomorphism the class of all lattices on n elements having nullity two, containing exactly four reducible elements.

In the era of big data, classical computing techniques face challenges in handling large and complex datasets. Quantum computing offers a transformative solution, especially in terms of real-time data processing speed. This study compares the performance of quantum and classical algorithms for large-scale data tasks. Results show that quantum algorithms achieve up to 70% faster processing and 30% greater computational efficiency, with scalability and an accuracy rate of 95% outperforming classical methods. Despite current limitations such as decoherence and error rates, ongoing advancements in quantum hardware and error correction highlight the potential of quantum computing to revolutionize data processing.

In this paper we introduce a natural mathematical structure derived from Samuel Beckett’s play “Quad”. We call this structure a binary Beckett-Gray code. We enumerate all codes for \(n \leq 6\) and give examples for \(n=7,8\). Beckett-Gray codes can be realized as successive states of a queue data structure. We show that the binary reflected Gray code can be realized as successive states of two stack data structures.