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.

Graph invariants, often regarded as topological indices, play a pivotal role in understanding and quantifying the structural properties of graphs. Among these, the line completion number has emerged as a significant measure of a graph’s edge connectivity and topology. In 1992, Bagga et al. defined a generalization of line graphs, namely super line graphs, and introduced the concept of the line completion number as a topological index of a graph. They calculated the line completion number for several classes of graphs, showcasing its utility in understanding graph structure. The line completion number of a graph, is the smallest index such that the super line graph becomes a complete graph. This index encapsulates the interplay between edge relationships and structural complexity, making it a versatile tool for characterizing graphs. Building upon this foundation, we analogously introduce the concepts of super point graphs and the point completion number, as vertex-centric topological indices. We establish a relationship between the point completion number and the line completion number, further extending the framework of graph invariants. Additionally, we compute the point completion numbers for various graph classes and analyze their structural implications. Our findings emphasize the significance of completion numbers as robust descriptors for graph topology, with potential applications in network analysis, chemistry, and other domains.