arXiv subjects

Muhammad Faisal Nadeem

Publications and source records attributed to Muhammad Faisal Nadeem.

4 recordsLinked to original sources

arXiv2026

A Graph Theoretical Approach to Optimizing Minimum Italian Domination Sets

A classical problem in graph theory known as the Italian domination number(also called Roman 2-domination number), involves assigning labels of 0, 1,or 2 to each node v. The goal is to ensure that every node with a label of 0 has a sum of labels of the nodes in its closed neighborhood that is 2 or greater. In computer systems, it is coined encompassing a robust cyber security strategy that will protect networks from potential threats, such as hacking, malware, and unauthorized access, by deploying security measures to provide the highest level of protection while reducing the misuse of resources. Toeplitz graphs are a special kind of graphs built over Toeplitz matrices from linear algebra, which are matrices with constant straight diagonal members. In this paper, we provide a detailed analysis regarding the Italian domination numbers for every Toeplitz graph family. We provide comprehensive results on Italian domination numbers across multiple graph families and identify the specific values at which the Italian domination number alters with increasing generator values.

math.CO↗

arXiv2025

Coreset selection based on Intra-class diversity

Deep Learning models have transformed various domains, including the healthcare sector, particularly biomedical image classification by learning intricate features and enabling accurate diagnostics pertaining to complex diseases. Recent studies have adopted two different approaches to train DL models: training from scratch and transfer learning. Both approaches demand substantial computational time and resources due to the involvement of massive datasets in model training. These computational demands are further increased due to the design-space exploration required for selecting optimal hyperparameters, which typically necessitates several training rounds. With the growing sizes of datasets, exploring solutions to this problem has recently gained the research community's attention. A plausible solution is to select a subset of the dataset for training and hyperparameter search. This subset, referred to as the corset, must be a representative set of the original dataset. A straightforward approach to selecting the coreset could be employing random sampling, albeit at the cost of compromising the representativeness of the original dataset. A critical limitation of random sampling is the bias towards the dominant classes in an imbalanced dataset. Even if the dataset has inter-class balance, this random sampling will not capture intra-class diversity. This study addresses this issue by introducing an intelligent, lightweight mechanism for coreset selection. Specifically, it proposes a method to extract intra-class diversity, forming per-class clusters that are utilized for the final sampling. We demonstrate the efficacy of the proposed methodology by conducting extensive classification experiments on a well-known biomedical imaging dataset. Results demonstrate that the proposed scheme outperforms the random sampling approach on several performance metrics for uniform conditions.

cs.CV↗

arXiv2025

Bounds on Atom-Bond Connectivity and Zagreb Indices in Trees with a Given Metric Dimension

Let $\mathbb{G} = (\mathcal{V}, \mathcal{E})$ be a simple connected graph, where $\mathcal{V}$ and $\mathcal{E}$ denote the vertex and edge sets, respectively. The first Zagreb index is defined as $\mathcal{M}_{1}(\mathbb{G}) = \sum_{v \in \mathcal{V}} ζ_{\mathbb{G}}(v)^2$, while the second Zagreb index is given by $\mathcal{M}_{2}(\mathbb{G}) = \sum_{uv \in \mathcal{E}} ζ_{\mathbb{G}}(u)\, ζ_{\mathbb{G}}(v)$, where $ζ_{\mathbb{G}}(v)$ represents the degree of vertex $v$. Another notable degree-based invariant is the atom-bond connectivity (ABC) index, introduced in chemical graph theory, and defined by \[ ABC(\mathbb{G}) = \sum_{uv \in \mathcal{E}} \sqrt{\frac{ζ_{\mathbb{G}}(u) + ζ_{\mathbb{G}}(v) - 2}{ζ_{\mathbb{G}}(u)\, ζ_{\mathbb{G}}(v)}}. \] A fundamental graph parameter, the metric dimension, refers to the minimum number of vertices in a resolving set that uniquely distinguishes all other vertices based on distances. In this work, we investigate the influence of metric dimension on the Zagreb and ABC indices within the class of trees. We derive sharp bounds-both upper and lower for $\mathcal{M}_1$ and $\mathcal{M}_2$, and provide an upper bound for the ABC index, all expressed in terms of the tree's order and its metric dimension. Furthermore, we identify the extremal tree structures that attain these bounds. These findings underscore the role of metric dimension in shaping topological descriptors and contribute both to theoretical graph analysis and practical applications in molecular chemistry.

math.GM↗

arXiv2024

Extremal Values of the Atom-Bond Connectivity Index for Trees with Given Roman Domination Numbers

Consider that $\mathbb{G}=(\mathbb{X}, \mathbb{Y})$ is a simple, connected graph with $\mathbb{X}$ as the vertex set and $\mathbb{Y}$ as the edge set. The atom-bond connectivity ($ABC$) index is a novel topological index that Estrada introduced in Estrada et al. (1998). It is defined as $$ A B C(\mathbb{G})=\sum_{xy \in Y(\mathbb{G})} \sqrt{\frac{ζ_x+ζ_y-2}{ζ_x ζ_y}} $$ where $ζ_x$ and $ζ_x$ represent the degrees of the vertices $x$ and $y$, respectively. In this work, we explore the behavior of the $A B C$ index for tree graphs. We establish both lower and upper bounds for the $A B C$ index, expressed in terms of the graph's order and its Roman domination number. Additionally, we characterize the tree structures that correspond to these extremal values, offering a deeper understanding of how the Roman domination number ($RDN$) influences the $A B C$ index in tree graphs.

math.GM↗