Shahid Rajaee Teacher Training UniversityJournal of Discrete Mathematics and Its Applications2981-08098420231201Vertex weighted Laplacian graph energy and other topological indices20120952410.22061/jdma.2023.524ENReza SharafdiniPersian Gulf University0000-0002-3171-2209Habibeh PanahbarDepartment of Mathematics, Faculty of Science, Persian Gulf University, Bushehr 7516913817,
I. R. IranJournal Article20231024Let $G$ be a graph with a vertex weight $omega$ and the vertices $v_1,ldots,v_n$. The Laplacian matrix of $G$ with respect to $omega$ is defined as $L_omega(G)=diag(omega(v_1),cdots,omega(v_n))-A(G)$, where $A(G)$ is the adjacency matrix of $G$. Let $mu_1,cdots,mu_n$ be eigenvalues of $L_omega(G)$. Then the Laplacian energy of $G$ with respect to $omega$ defined as $LE_omega (G)=sum_{i=1}^nbig|mu_i - overline{omega}big|$, where $overline{omega}$ is the average of $omega$, i.e., $overline{omega}=dfrac{sum_{i=1}^{n}omega(v_i)}{n}$. In this paper we consider several natural vertex weights of $G$ and obtain some inequalities between the ordinary and Laplacian energies of $G$ with corresponding vertex weights. Finally, we apply our results to the molecular graph of toroidal fullerenes (or achiral polyhex nanotorus).\[5mm] noindenttextbf{Key words:} Energy of graph, Laplacian energy, Vertex weight, Topological index, toroidal fullerenes.https://jdma.sru.ac.ir/article_524_cec3cdf0d8580a284fcf20814e91c1d8.pdfShahid Rajaee Teacher Training UniversityJournal of Discrete Mathematics and Its Applications2981-08098420231201Eyes on the cosmic web: A tribute to Ali Reza Ashrafi211224208710.22061/jdma.2023.2087ENOttorino OriActinium Chemical Research, Via Casilina 1626/A, 00133, Rome, ItalyMihai PutzLaboratory of Computational and Structural Physical-Chemistry for Nanosciences and QSAR, Biology-Chemistry Department, Faculty of Chemistry, Biology, Geography, West University of Timisoara, Pestalozzi Str. No. 16A,RO-300115 Timisoara, RomaniaJournal Article20231005This article delves into the subject of topological modeling and invariants in the study of the Cosmic Web (CW), which refers to the vast network of galaxies in the universe. The article explores the use of eccentric connectivity and other topological descriptors to classify various structures within the Cosmic Web, such as filaments, walls, and clusters. By analyzing graphs and lattices in detail, the study shows how topological invariants can be used to extract morphological information and categorize nodes based on their structural roles. Additionally, the article discusses the potential application of these methods in assigning galaxy populations across different structures within the Cosmic Web. This research offers valuable insights into the use of topological tools for comprehending the intricate and complex nature of the universe's large-scale galaxy distribution.https://jdma.sru.ac.ir/article_2087_118656eb80089d8175e911bdafc9661f.pdfShahid Rajaee Teacher Training UniversityJournal of Discrete Mathematics and Its Applications2981-08098420231201Computing Degree-Based Topological Indices of Polyhex Nanotubes22523352510.22061/jdma.2023.526ENVijayalaxmi ShigehalliRani Channamma University, Belagavi-591156, Karnataka, India.Rachanna KanaburRANI CHANNAMMA University, BELAGAVI-5911560000000174967503Journal Article20231101Recently, Shigehalli and Kanabur [20] have put forward for new degree based topological indices, namely Arithmetic-Geometric index (AG1 index), SK index, SK<sub>1</sub> index and SK<sub>2</sub> index of a molecular graph G. In this paper, we obtain the explicit formulae of these indices for Polyhex Nanotube without the aid of a computer.https://jdma.sru.ac.ir/article_525_506baf440bd11a09d1e52c22b4f13853.pdfShahid Rajaee Teacher Training UniversityJournal of Discrete Mathematics and Its Applications2981-08098420231201On the conjecture for the sum of the largest signless Laplacian eigenvalues of a graph- a survey235245202610.22061/jdma.2023.10290.1061ENShariefuddin PirzadaDepartment of Mathematics, University of Kashmir, IndiaJournal Article20231027Let $G$ be a simple graph with order $n$ and size $m$. Let $D(G)=$ diag$(d_1, d_2, \dots, d_n)$ be its diagonal matrix, where $d_i=\deg(v_i),$ for all $i=1,2,\dots,n$ and $A(G)$ be its adjacency matrix. The matrix $Q(G)=D(G)+A(G)$ is called the signless Laplacian matrix of $G$. Let $q_1,q_2,\dots,q_n$ be the signless Laplacian eigenvalues of $Q(G)$ and let $S^{+}_{k}(G)=\sum_{i=1}^{k}q_i$ be the sum of the $k$ largest signless Laplacian eigenvalues. Ashraf et al. [F. Ashraf, G. R. Omidi, B. Tayfeh-Rezaie, On the sum of signless Laplacian eigenvalues of a graph, Linear Algebra Appl. {\bf 438} (2013) 4539-4546.] conjectured that $S^{+}_{k}(G)\leq m+{k+1 \choose 2}$, for all $k=1,2,\dots,n$. We present a survey about the developments of this conjecture.https://jdma.sru.ac.ir/article_2026_01246c750b33bbcb11ff7aeadf5fd59e.pdfShahid Rajaee Teacher Training UniversityJournal of Discrete Mathematics and Its Applications2981-08098420231201Ramanujan Cayley graphs on sporadic groups247261202810.22061/jdma.2023.10294.1062ENShahram MehryDepartment of Mathematics, Faculty of Science, Bu ali sina University,
Hamedan, IranJournal Article20231101Let $\Gamma$ be a $k-$regular graph with the second maximum eigenvalue $\lambda$. Then $\Gamma$ is said o be Ramanujan graph if $\lambda\leq 2\sqrt{k-1}.$<br /> Let $G$ be a finite group and $\Gamma=Cay(G,S)$ be a Cayley graph related to $G$. <br /> The aim of this paper is to investigate the Ramanujan Cayley graphs of<br /> sporadic groups.https://jdma.sru.ac.ir/article_2028_c506679c585cfc640b7839e33753b664.pdfShahid Rajaee Teacher Training UniversityJournal of Discrete Mathematics and Its Applications2981-08098420231201ON PAIRS OF NON-ABELIAN FINITE P-GROUPS210910.22061/jdma.2024.10706.1068ENElaheh KhamsehDepartment of Mathematics, Islamic Azad University, Shahr-e-Qods Branch, Tehran, Iran.Journal Article20240228Let (N;G) be a pair of non-abelian finite p-groups and K be a<br />normal subgroup of G such that G = N \times K, where K is a d-generator group<br />of order pm. Moreover, let |N| = p^n and {N'| = p^k. Then |M(N;G)|=<br />p^{1/2 (n-1)(n-2)+1+(n-1)m-s'}<br />, where M(N;G) is the Schur multiplier of the<br />pair (N;G) and s0 is a non-negative integer. In this paper, the non-abelian<br />pairs (N;G) for s0 = 0; 1; 2; 3 are characterized.