Journal of Discrete Mathematics and Its Applications
https://jdma.sru.ac.ir/
Journal of Discrete Mathematics and Its Applicationsendaily1Fri, 01 Dec 2023 00:00:00 +0330Fri, 01 Dec 2023 00:00:00 +0330Vertex weighted Laplacian graph energy and other topological indices
https://jdma.sru.ac.ir/article_524.html
Let $G$ 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&nbsp;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.Eyes on the cosmic web: A tribute to Ali Reza Ashrafi
https://jdma.sru.ac.ir/article_2087.html
This 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.Computing Degree-Based Topological Indices of Polyhex Nanotubes
https://jdma.sru.ac.ir/article_525.html
Recently, Shigehalli and Kanabur [20] have put forward for new degree based topological indices, namely Arithmetic-Geometric index (AG1 index), SK index, SK1 index and SK2 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.On the conjecture for the sum of the largest signless Laplacian eigenvalues of a graph- a survey
https://jdma.sru.ac.ir/article_2026.html
Let $G$ 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.Ramanujan Cayley graphs on sporadic groups
https://jdma.sru.ac.ir/article_2028.html
Let $\Gamma$ be a $k-$regular graph with the second maximum &nbsp;eigenvalue $\lambda$. Then &nbsp;$\Gamma$ is said o be Ramanujan graph if $\lambda\leq 2\sqrt{k-1}.$&nbsp;Let $G$ be a finite group &nbsp;and $\Gamma=Cay(G,S)$ be a Cayley graph related to $G$.&nbsp;&nbsp;The aim of this paper is to investigate the Ramanujan Cayley graphs of&nbsp;sporadic groups.ON PAIRS OF NON-ABELIAN FINITE P-GROUPS
https://jdma.sru.ac.ir/article_2109.html
Let (N;G) be a pair of non-abelian finite p-groups and K be anormal subgroup of G such that G = N \times K, where K is a d-generator groupof order pm. Moreover, let |N| = p^n and {N'| = p^k. Then |M(N;G)|=p^{1/2 (n-1)(n-2)+1+(n-1)m-s'}, where M(N;G) is the Schur multiplier of thepair (N;G) and s0 is a non-negative integer. In this paper, the non-abelianpairs (N;G) for s0 = 0; 1; 2; 3 are characterized.