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.

In IoT-managed power systems, equipment or communication failures can result in missing or abnormal power quality data, making data restoration increasingly important. Traditional repair methods often struggle to capture complex data relationships and suffer from low accuracy. This paper proposes a power quality data restoration approach based on a low-rank matrix completion algorithm to enhance repair accuracy and efficiency. The system consists of three main steps: data preprocessing, matrix completion, and result validation. Z-score normalization is applied to raw data, and Singular Value Decomposition (SVD) is used for low-rank approximation in matrix filling. Cross-validation and error metrics are employed to assess performance. Experimental results show that at a 10% missing rate, the mean square error is approximately 0.1. The proposed method demonstrates superior performance over traditional approaches, particularly at low missing rates, offering reliable support for monitoring and control in power IoT systems.

The current changes in China’s population structure and dynamics have led to profound challenges in population planning, forecasting, decision-making, and early warning. To address the issues of predicting age- and gender-specific population retention, migration, and birth rates, a combination model of Multilayer Perceptron (MLP) and Random Forest (RF) is constructed using stacking techniques, with a discrete population development equation as the base model. The MLP-RF model is employed to perform regression training on population data, resulting in a novel ensemble approach to population forecasting. The study uses the data from the sixth and seventh national censuses of Hebei Province, reconstructing population data for 2010-2020. After data training and error evaluation, it is demonstrated that the ensemble forecasting model has excellent predictive capabilities for population retention, migration, and birth-related issues.

In the current energy-constrained era, promoting electric vehicles (EVs) is a necessary trend. However, the simultaneous and uncoordinated charging of diverse EVs can negatively impact the power grid. This paper proposes a scaled EV orderly scheduling model, comprising charging demand simulation and a scheduling algorithm. Monte Carlo simulation, based on charging probability models, is used to generate EV cluster entry information and preprocess parameters. Two control strategies are proposed for clean energy dispatch and EV-based grid operation, accounting for user behavior-induced load variations. A microgrid optimization model is developed, with economic cost weights calculated. The model is solved using an improved PSO algorithm (APSO). Results show the APSO achieves better performance, with hourly average exchange loads of 2.7092 P/kW (vs. 1.9979 P/kW for PSO). Under 30–80% user responsiveness, microgrid management and environmental costs are reduced to 28,618.439 yuan and 7,864.685 yuan, respectively.

This paper investigates human-computer communication within the framework of deep learning and identifies three key features of such interaction. A cross-cultural empathy feature aliasing model based on Graph Neural Network-Attention Mechanism-Bi-directional Gating Unit (GCN-Attention-BiGRU) is proposed, with categorical cross-entropy and L2 regularization as the loss function. By integrating IoT and deep learning, an adaptive interaction model is developed and evaluated through experiments. Results show high mean scores for empathy (4.537), relevance (4.447), and fluency (4.499) across 60 samples, indicating effective empathy feature extraction. Additionally, the proposed model demonstrates greater efficiency and adaptability compared to traditional interaction models, enhancing cross-cultural empathy in human-computer communication.