arXiv subjects

Fouzul Atik

Publications and source records attributed to Fouzul Atik.

8 recordsLinked to original sources

arXiv2025

On the Spectral Analysis of the Superpower Graph of the Direct Product of Dihedral Groups

The superpower graph of a finite group $G$, or $\mathcal{S}_G$, is an undirected simple graph whose vertices are the elements of the group $G$, and two distinct vertices $a,b\in G$ are adjacent if and only if the order of one vertex divides the order of the other vertex, which means that either $o(a)|o(b)$ or $o(b)|o(a)$. In this paper, we have investigated the $A_\alpha$-adjacency spectral properties of the superpower graph of the direct product $D_p\times D_p$, where $D_p$ is a dihedral group for $p$ being prime. Also, we have determined its Laplacian and signless Laplacian spectrum by giving different values to $\alpha$; furthermore, we delved into its superpower graph and deduced the $A_\alpha$- adjacency spectrum of the superpower graph of $D_p\times D_p$ and $D_{p^m}$ for $p$ being an odd prime.

math.SP↗

arXiv2025

When Are Standard Graph Products Isomorphic?

This article investigates the isomorphism problem for graphs derived from the four standard graph products: Cartesian, Kronecker (direct), strong, and lexicographic product. We provide a complete characterization of all simple connected graphs for which their corresponding products are isomorphic. As a by-product, we identify a novel family of non-distance-regular graphs that possess fewer than d+1 distinct distance eigenvalues, where d represents the diameter of the graph. This result offers a new perspective on Problem 4.3 posed in [2], moving beyond the current approaches.

math.CO↗

arXiv2025

On the Spectral Analysis of Power Graph of Dihedral Groups

The power graph $ \mathcal{G}_G $ of a group $ G $ is a graph whose vertex set is $ G $, and two elements $ x, y \in G $ are adjacent if one is an integral power of the other. In this paper, we determine the adjacency, Laplacian, and signless Laplacian spectra of the power graph of the dihedral group $ D_{2pq} $, where $ p $ and $ q $ are distinct primes. Our findings demonstrate that the results of Romdhini et al. [2024], published in the \textit{European Journal of Pure and Applied Mathematics}, do not hold universally for all $ n \geq 3 $. Our analysis demonstrates that their results hold true exclusively when $ n = p^m $ where $ p $ is a prime number and $ m $ is a positive integer. The research examines their methodology via explicit counterexamples to expose its boundaries and establish corrected results. This study improves past research by expanding the spectrum evaluation of power graphs linked to dihedral groups.

math.SP↗

arXiv2024

On the distance spectrum of the Kronecker product of distance regular graphs

Consider two simple graphs, G1 and G2, with their respective vertex sets V(G1) and V(G2). The Kronecker product forms a new graph with a vertex set V(G1) X V(G2). In this new graph, two vertices, (x, y) and (u, v), are adjacent if and only if xu is an edge in G1 and yv is an edge in G2. While the adjacency spectrum of this product is known, the distance spectrum remains unexplored. This article determines the distance spectrum of the Kronecker product for a few families of distance regular graphs. We find the exact polynomial, which expresses the distance matrix D as a polynomial of the adjacency matrix, for two distance regular graphs, Johnson and Hamming graphs. Additionally, we present families of distance integral graphs, shedding light on a previously posted open problem given by Indulal and Balakrishnan in (AKCE International Journal of Graphs and Combinatorics, 13(3); 230 to 234, 2016).

math.SP↗

arXiv2024

Spectral radius and second largest eigenvalues of power graphs of finite groups

Consider a group $\mathbb{G}$ and construct its power graph, whose vertex set consists of the elements of $\mathbb{G}$. Two distinct vertices (elements) are adjacent in the graph if and only if one element can be expressed as an integral power of the other. In this article, we improved the bounds of the spectral radius of the power graphs of the cyclic group $C_{n}$, the dihedral group $\mathcal{D}_{2n}$, and the dicyclic group $\mathcal{Q}_{4n}$. For $n\neq p^{m},$ the power graph of the cyclic group $C_{n}$ is not a complete multipartite graph. We find the second largest eigenvalue bounds of the same with the clique number. In some cases, we find the bounds are exact if and only if they belong to a particular family of graphs. Lastly, we work on the distance spectral radius of the power graphs of the same groups

math.SP↗

arXiv2021

Analysis of Distributed Average Consensus Algorithms for Robust IoT networks

Internet of Things(IoT) is a heterogeneous network consists of various physical objects such as large number of sensors, actuators, RFID tags, smart devices, and servers connected to the internet. IoT networks have potential applications in healthcare, transportation, smart home, and automotive industries. To realize the IoT applications, all these devices need to be dynamically cooperated and utilize their resources effectively in a distributed fashion. Consensus algorithms have attracted much research attention in recent years due to their simple execution, robustness to topology changes, and distributed philosophy. These algorithms are extensively utilized for synchronization, resource allocation, and security in IoT networks. Performance of the distributed consensus algorithms can be effectively quantified by the Convergence Time, Network Coherence, Maximum Communication Time-Delay. In this work, we model the IoT network as a q-triangular r-regular ring network as q-triangular topologies exhibit both small-world and scale-free features. Scale-free and small-world topologies widely applied for modelling IoT as these topologies are effectively resilient to random attacks. In this paper, we derive explicit expressions for all eigenvalues of Laplacian matrix for q-triangular r-regular networks. We then apply the obtained eigenvalues to determine the convergence time, network coherence, and maximum communication timedelay. Our analytical results indicate that the effects of noise and communication delay on the consensus process are negligible for q-triangular r-regular networks. We argue that q-triangulation operation is responsible for the strong robustness with respect to noise and communication time-delay in the proposed network topologies.

cs.DC↗

arXiv2018

Resistance matrices of graphs with matrix weights

The \emph{resistance matrix} of a simple connected graph $G$ is denoted by $R$, and is defined by $R =(r_{ij})$, where $r_{ij}$ is the resistance distance between the vertices $i$ and $j$ of $G$. In this paper, we consider the resistance matrix of weighted graph with edge weights being positive definite matrices of same size. We derive a formula for the determinant and the inverse of the resistance matrix. Then, we establish an interlacing inequality for the eigenvalues of resistance and Laplacian matrices. Using this interlacing inequality, we obtain the inertia of the resistance matrix.

math.CO↗

arXiv2017

On distance and Laplacian matrices of trees with matrix weights

The \emph{distance matrix} of a simple connected graph $G$ is $D(G)=(d_{ij})$, where $d_{ij}$ is the distance between the vertices $i$ and $j$ in $G$. We consider a weighted tree $T$ on $n$ vertices with edge weights are square matrix of same size. The distance $d_{ij}$ between the vertices $i$ and $j$ is the sum of the weight matrices of the edges in the unique path from $i$ to $j$. In this article we establish a characterization for the trees in terms of rank of (matrix) weighted Laplacian matrix associated with it. Then we establish a necessary and sufficient condition for the distance matrix $D$, with matrix weights, to be invertible and the formula for the inverse of $D$, if it exists. Also we study some of the properties of the distance matrices of matrix weighted trees in connection with the Laplacian matrices, g-inverses and eigenvalues.

math.CO↗