Mean coloring is an edge coloring c of a connected graph G of order 3 or more with positive integers if the chromatic mean of all vertices v of G are integers. The chromatic mean of a vertex v of G is given by cm(v)=∑ _e ∈ E_v c(e)/deg(v) where c(e) is the integers(colors) given to the edges incident to the vertex v. If each vertex has a distinct chromatic mean, then the edge coloring c is called a rainbow mean coloring. For a rainbow mean coloring c of a graph G, the maximum vertex color is the rainbow chromatic mean index, rm(c) of c. The rainbow mean index rm(G) of the graph G is defined as rm(G)=min {rm(c): c is the rainbow mean coloring} . In this paper, we determine the rainbow mean index of bistars and tadpole graphs. Also, we show that the rainbow mean index is same as the order for some derived graphs.
We present the 2024 edition of the ECN AI Baseline Index (EAII), an accessible benchmark evaluating large language models in competitive programming under contest conditions. The universities division of the 2024 Sapientia–ECN (Efficiency, Challenge, Networking) programming contest included 11 teams and 16 university-track problems and introduced a judging platform restricted to C/C++/Pascal. Using a simple C++ prompting protocol with limited feedback rounds, the AI fully solved 13 of 16 problems (81.25
We study the numerical computation of nontrivial critical points of variational functionals associated with nonlinear Dirichlet problems involving the p-Laplacian. Previous numerical mountain pass approaches typically relied on finite element discretizations of the underlying function space. In contrast, we employ a discretization based on a geometric B-spline representation of the solution. The function space is approximated by smooth spline curves parameterized by control points, yielding a finite-dimensional geometric representation of the variational problem. Within this discrete space we apply a mountain-pass type up–down method. This allows the search for saddle-type critical points to be carried out directly in the space of spline control points. The descent direction is obtained through an auxiliary Poisson equation, providing a Sobolev gradient that stabilizes the iteration. Convergence of the numerical procedure is monitored via the Euler–Lagrange residual, ensuring that the computed spline approximation satisfies the variational problem up to a prescribed tolerance. Numerical experiments for the model case p=2 with nonlinearity f(u)=u^3 on Ω =(0,1) show that the method computes nontrivial solutions, including sign-changing profiles depending on the initialization. The results demonstrate the successful application of B-splines in the numerical solution of nonlinear variational problems.
We present a heuristic for the error in the output of holographic associative memory while using the basic training regime, and propose a new formula for the normalisation coefficient used during recalling. We conduct experiments to measure the adequacy of our heuristic, and also to look at the effect of symmetrisation and expansion, while using mainly visual stimuli with responses encoded as codewords of Reed–Muller codes.
Recent advances in natural language processing (NLP) have enabled sophisticated analyses of textual data, including the lyrics of popular music. This study applies statistical NLP methods to a large corpus of Hungarian popular music lyrics to examine differences and similarities across 10 musical genres. We investigate whether genre classification can be achieved using lyrics alone through multilabel and pairwise classification experiments, employing multinomial naive Bayes and huBERT models. In addition, we assess textual complexity using two approaches, namely the Flesch–Kincaid formula and gzip compression ratio, exploring potential correlations with genre popularity. Sentiment analysis further provides a positivity ranking of genres, allowing us to confirm or challenge common assumptions about their thematic characteristics, such as the prevalence of negative sentiment in metal music. The findings are compared with prior studies on English-language lyrics, offering insights into cross-linguistic and cultural patterns in musical expression, and we hope that this work will pave the way for further similar research. The primary objective of the present study is to demonstrate the feasibility of constructing corpora of lyrical texts for languages that are not widely utilized, and of concomitantly accumulating metadata for these corpora. The overarching ambition of these endeavors is to enable the conduct of an extensive array of experiments, thereby advancing research in the field.
Global feature importance (GFI) methods are essential tools for interpreting machine learning models, yet their application in more complex forecasting tasks involving multiple time series can be challenging. This study focuses on tree-based ensemble models applied to multi-series product demand forecasting. To evaluate the different GFI methods, using controlled simulated datasets with controlled dependencies on lagged values and external demand drivers, we compare model-specific and model-agnostic global importance methods, including Shapley values, permutation importance, and tree-specific gain- and split-based importance. Our analysis focuses on uncovering pitfalls in applying these methods, including problems introduced by auto-correlation and feature scaling. This work contributes practical guidance for practitioners seeking to apply these methods in real-world forecasting scenarios and to leverage explainability methods for informed decision-making.
A set D ⊆ V of a graph G =(V, E) is called a dominating set of G if every vertex in V∖ D is adjacent to at least one vertex in D. A dominating set D of a graph G is convex dominating set if all vertices from u-v geodesic belong to D for every two vertices u,v ∈ D . A convex dominating set D of a graph G is nonsplit convex dominating set if the induced subgraph G[V ∖ D] is connected. The nonsplit convex domination number of G is the minimum cardinality of a nonsplit convex dominating set D and it is denoted by γ _nscon(G) . In this paper, we initiate the study on this parameter. We establish bounds for nonsplit convex domination number, γ _nscon(G) , of standard graph structures. Further, we also present conditions for identifying or constructing a nonsplit convex dominating set in any connected graph G.
Academic research workflows can be significantly accelerated by tools that provide targeted access to the most relevant parts of scientific articles, avoiding the need to read full documents. Since interest typically centres on key elements such as the research problem, proposed approach, and main findings, automatic detection of a paper’s structural sections can greatly facilitate navigation and content retrieval. In this work, we address the task of sentence-level section classification in research papers, assigning each sentence to categories such as Introduction, Related Work, Proposed Approach, Body, Results, and Conclusion. We propose a classification pipeline built on a high-quality dataset constructed via a dedicated filtering and selection process. Two pre-trained models are first used to automatically label sentences extracted from research articles; only sentences for which both models agree on the predicted label are retained, ensuring consistent annotation. On this refined dataset, we fine-tune a BERT-Base* classifier that achieves an average precision of 84.67
Let G be a simple connected simple graph of order n. The distance Laplacian matrix D^L(G) is defined as D^L(G)=Diag(Tr)-D(G) , where Diag(Tr) is the diagonal matrix of vertex transmissions and D(G) is the distance matrix of G. The eigenvalues of D^L(G) are the distance Laplacian eigenvalues of G and are denoted by ∂ _1^L(G), ∂ _2^L(G),… ,∂ _n^L(G) . The distance Laplacian spread DLS(G) of a connected graph G is the difference between largest and second smallest distance Laplacian eigenvalues, that is, ∂ _1^L(G)-∂ _n-1^L(G) . We obtain bounds for DLS(G) in terms of the Wiener index W(G), order n and the maximum transmission degree Tr_max(G) of G and characterize the extremal graphs. We obtain two lower bounds for DLS(G), the first one in terms of the order, diameter and the Wiener index of the graph, and the second one in terms of the order, maximum degree and the independence number of the graph. For a connected k-partite graph G, k≤ n-1 , with n vertices having disconnected complement, we show that DLS(G)≥⌊n/k⌋ with equality if and only if G is a complete balanced k-partite graph.
In this paper we introduce and study the diminished Sombor index and the associated diminished Sombor spectrum of the comaximal graph Γ (ℤ_n) of the ring of integers modulo n. The diminished Sombor matrix is defined by replacing each nonzero off-diagonal entry of the adjacency matrix with the normalised weight √(d(u)^2+d(v)^2)/d(u)+d(v), where d(u) and d(v) denote the degrees of the adjacent vertices u, v. We establish general bounds for the diminished Sombor index DSO(Γ (ℤ_n)) , characterise the extremal cases, and describe in detail the diminished Sombor spectrum of Γ (ℤ_n) . In particular, we show that the eigenvalue -1/√(2) occurs with multiplicity φ (n)-1 , the eigenvalue 0 occurs with multiplicity n-φ (n)-1-t , and that the remaining t+2 eigenvalues are precisely the eigenvalues of a reduced quotient matrix depending only on n and its prime-power divisors. Using this spectral description we derive explicit formulas and bounds for the diminished Sombor energy of Γ (ℤ_n) . We analyzed certain special cases viz. prime, prime power, and product of two or three primes in detail, and illustrative numerical examples are provided. Finally, we conclude the paper with some open problems and directions for future research.
Graphs with an ideally restricted spectrum are known as Ramanujan graphs. In computer science and combinatorics, these graphs have several uses. This paper explores the metric dimension of Ramanujan graphs. Metric dimension is the minimum “Locator points” needed to uniquely locate every spot in a network. Ramanujan graphs are highly efficient and well-connected networks. Our research investigates how their unique mathematical properties influence this “Locator points” requirement, a problem typically very complex. By using their inherent design, this work aims to understand these networks better and show their potential for precise location-finding in large systems. General codes for these graphs have also been discussed.
There are several mutually exclusive views on Software Engineering: empirical studies suggest the existence of natural laws that govern development, companies promote tools and methods promising groundbreaking results, and developers describe it as a creative and innovative endeavour. In this study, we significantly expand our previous research on software evolution, performing a longitudinal analysis of 65, 987 popular open-source projects on GitHub to investigate project trajectories. We also reflect on Les Hatton’s observation that the emergence of some properties might be “divorced from human agency”. We examined projects written in 85 different languages and found a separation based on the volume of commits. Projects with ≥ approx. 700 commits to their main branch (10, 612, or 16.1% of the total) exhibit trends consistent with highly automated workflows. These large projects have been resilient to external events over the last few decades, show a higher likelihood of accelerating production, and deterministic evolution patterns, regardless of when they were started or how long they were developed. In contrast, the vast majority of smaller projects tend to exhibit less deterministic evolution patterns and are significantly more likely to experience deceleration over time. These findings reveal a separation in how different software systems evolve that might have significant implications for how we understand, study, and teach programming. Specifically, in our interpretation, the observed numerical dominance of smaller projects, potentially following less regular workflows, suggests a need for data curation to prevent the risk of assimilating and disseminating practices based not on their evolutionary success or long-term sustainability, but rather on their statistical prevalence among the numerous, potentially less mature projects. Researchers must also note that focusing solely on large projects can overlook a much larger and different set of projects that could benefit from a targeted study.
Interactions in multi-agent systems are often framed through the tools of game theory; however, in real-world scenarios, the structure and parameters of the underlying game faced by agents are frequently unknown or non-stationary. This presents a critical challenge: agents must rapidly infer the nature of their environment and adapt their strategies accordingly, even in the presence of multiple other agents. Meta-reinforcement learning (meta-RL) has demonstrated the ability to facilitate fast adaptation in tasks such as multi-armed bandits, Markov decision processes, and visual navigation. In this paper, we extend the application of meta-RL to multi-agent games. By training agents via self-play meta-reinforcement learning on diverse classes of normal-form games, parameterized by their payoff matrices and sampled from a distribution, we develop algorithms that are not only sample-efficient and robust to changes, but also capable of strategic generalization across distinct game-theoretic structures. Although it remains limited to a theoretical proof of concept, our approach bridges the gap between classical game-theoretic modeling and modern meta-learning techniques, with promising implications for adaptive behavior in dynamic multi-agent environments.
This paper explores the newly introduced concept of paired disjunctive domination, initially proposed by Henning et al. A subset D ⊆ V is called a disjunctive dominating set of a graph G for each vertex v ∈ V , if there exists either a vertex in D adjacent to v , or at least two vertices in D whose distance from v is exactly two in G . Additionally, a disjunctive dominating set D ⊆ V is defined as a paired disjunctive dominating set if the induced subgraph by D in G contains a perfect matching. In this work, we present new results concerning the R-vertex, R-edge, R-vertex neighborhood, and R-edge neighborhood corona structures based on this parameter.
Centrality measures are used to quantify the influence or importance of vertices within a graph. The stress of an internal vertex u in a connected graph G is a centrality measure defined as the number of shortest paths passing through a vertex u in a graph G. In this paper, we introduce a new vertex-based invariant, complementing to stress, called relief of a vertex u in a connected graph G denoted by Re_G(u) and is defined as the number of shortest paths not passing through a vertex u in a graph G. The sum of relief of all vertices of a connected graph G is called relief of G denoted by Re(G). We obtain the relief of a vertex in a tree and hence obtain the relief of trees. We construct a tree where relief and stress of a vertex is the same. Further linear regression analysis of the relief with the physico-chemical properties of alkanes is carried out. The linear model, based on the relief shows good correlation with the physico-chemical properties of alkanes.
For a family of graphs ℋ , a graph G is ℋ -free if it does not contain a subgraph isomorphic to any graph in ℋ . Nikiforov (Linear Algebra Appl 432:2243–2256, 2010) proposed a spectral analogue of the Turán problem which asks to determine the maximum spectral radius of an H-free graph of size m or order n. Let F_k be the graph obtained from k intersecting triangles sharing a common vertex. For k=2 , the graph F_2 is called the bowtie. Let S_n,2 be the graph obtained by joining each vertex of K_2 to n-2 isolated vertices and let S^-_n,2 be the graph obtained from S_n,2 by deleting an edge incident to vertex of degree 2. Li et al. (Discrete Math 346:113680, 2023) showed that ρ (G)≤1+√(4m-3)/2 for any F_2 -free graph of size m≥ 8 . They also proved that the unique extremal graph is the join of K_2 with an independent set of m-1/2 vertices. However, this bound is only sharp for odd m. Let ℱ be a family of graphs, where any m-edge graph G in ℱ is obtained from a bipartite graph B(S, W) (not necessarily complete) by adding two new vertices u and v_0 , an edge uv_0 , all edges between u and S, and t edges between v_0 and S, where t≤ |N(u)| . In this paper, we determine a sharp upper bound for ρ (G) in F_2 -free graphs that do not belong to ℱ , given that G has m (even) edges.
Time series classification, a key area within artificial intelligence and machine learning, has widespread applications across economics, finance, medicine, and engineering. Such applications include verifying signatures on touchscreens, recognizing physical activities using accelerometer data, identifying users through keystroke dynamics, and many more. This paper provides an overview of the most significant time series classification methods developed over the last two decades. These methods range from nearest neighbor models with dynamic time warping, to deep learning techniques like convolutional and residual networks (ResNet), and more recent approaches such as the Random Convolutional Kernel Transform (ROCKET). A notable insight from this review is that relatively simple models often perform remarkably well for time series classification tasks.
Virtual assistants (VAs) have become essential instruments in modern technology, transforming human–machine interactions across various disciplines. Their use of natural language processing, cloud, and IoT has facilitated assistance with activities from everyday conveniences to intricate decision-making in healthcare, education, and other domains. This review seeks to consolidate disparate research by offering a novel comprehensive taxonomy of virtual assistants, as well as smart speakers, as a key embodiment of VAs, highlighting their integration with IoT ecosystems and their ability to enable intuitive, multimodal interactions. This study employs a thorough literature analysis to assess diverse applications of VAs across domains, while also critically examining the challenges and limitations. By addressing these barriers and consolidating fragmented research, this study provides a unified framework for understanding VAs and their future role in fostering intelligent, inclusive ecosystems.
Neuroblastoma is a complex pediatric cancer with high molecular heterogeneity, making accurate subtyping and prognosis prediction challenging. Traditional clustering methods struggle with the high dimensionality of multi-omics data, leading to suboptimal stratification of diseses. This study presents a deep learning-assisted multi-objective clustering framework integrating autoencoders for dimensionality reduction with the Non-Dominated Sorting Genetic Algorithm II (NSGA-II) to enhance clustering accuracy and biological relevance. The autoencoder extracts essential features while minimizing noise, generating a compact latent representation of multi-omics data. NSGA-II then optimizes intra-cluster compactness and inter-cluster separation to improve patient stratification. The framework was validated on publicly available neuroblastoma datasets encompassing genomics, transcriptomics, and epigenomics data. Comparative analysis showed that our approach improves clustering accuracy by 15
Accurate insect species identification is vital for pest monitoring and biodiversity assessment but traditionally demands taxonomic expertise and significant time investment. Low-code AI platforms offer accessible solutions for automating image classification, enabling non-experts to leverage machine learning. This study evaluates two low-code AI platforms—Google Cloud AutoML Vision and Google Teachable Machine—for classifying eight insect species from the Cholistan Desert, Pakistan. A dataset of 9600 images, collected between March and October 2023, was divided into training (80
