On divisibility graph for simple Zassenhaus groups

The divisibility graph $D(G)$ for a finite group $G$ is a graph with vertex set $cs~(G)\setminus\{1\}$ where $cs~(G)$ is the set of conjugacy class sizes of $G$. Two vertices $a$ and $b$ are adjacent whenever $a$ divides $b$ or $b$ divides $a$. In this paper we will find $D(G)$ where $G$ is a simple Zassenhaus group

Divisibility graph for symmetric and alternating groups

Let $X$ be a non-empty set of positive integers and $X^*=X\setminus \{1\}$. The divisibility graph $D(X)$ has $X^*$ as the vertex set and there is an edge connecting $a$ and $b$ with $a, b\in X^*$ whenever $a$ divides $b$ or $b$ divides $a$. Let $X=cs~{G}$ be the set of conjugacy class sizes of a group $G$. In this case, we denote $D(cs~{G})$ by $D(G)$. In this paper we will find the number of connected components of $D(G)$ where $G$ is the symmetric group $S_n$ or is the alternating group $A_n$

Quotient graphs for power graphs

In a previous paper of the first author a procedure was developed for counting the components of a graph through the knowledge of the components of its quotient graphs. We apply here that procedure to the proper power graph $\mathcal{P}_0(G)$ of a finite group $G$, finding a formula for the number $c(\mathcal{P}_0(G))$ of its components which is particularly illuminative when $G\leq S_n$ is a fusion controlled permutation group. We make use of the proper quotient power graph $\widetilde{\mathcal{P}}_0(G)$, the proper order graph $\mathcal{O}_0(G)$ and the proper type graph $\mathcal{T}_0(G)$. We show that all those graphs are quotient of $\mathcal{P}_0(G)$ and demonstrate a strong link between them dealing with $G=S_n$. We find simultaneously $c(\mathcal{P}_0(S_n))$ as well as the number of components of $\widetilde{\mathcal{P}}_0(S_n)$, $\mathcal{O}_0(S_n)$ and $\mathcal{T}_0(S_n)$

The Divisibility Graph of finite groups of Lie Type

The Divisibility Graph of a finite group $G$ has vertex set the set of conjugacy class lengths of non-central elements in $G$ and two vertices are connected by an edge if one divides the other. We determine the connected components of the Divisibility Graph of the finite groups of Lie type in odd characteristic

Applications of blockchain technology in sustainable manufacturing and supply chain management: A systematic review

Developing sustainable products and processes is essential for the survival of manufacturers in the current competitive market and the industry 4.0 era. The activities of manufacturers and their supply chain partners should be aligned with sustainable development goals. Manufacturers have faced many barriers and challenges in implementing sustainable practices along the entire supply chain due to globalisation, outsourcing, and offshoring. Blockchain technology has the potential to address the challenges of sustainability. This study aims to explain the applications of blockchain technology to sustainable manufacturing. We conducted a systematic literature review and explained the potential contributions of blockchain technology to the economic, environmental, and social performances of manufacturers and their supply chains. The findings of the study extend our understanding of the blockchain applications in sustainable manufacturing and sustainable supply chains. Furthermore, the study explains how blockchain can influence the sustainable performance of manufacturers by creating transparency, traceability, real-time information sharing, and security of the data capabilities

On Harmonic Index and Diameter of Quasi-Tree Graphs

The harmonic index of a graph G (HG) is defined as the sum of the weights 2/du+dv for all edges uv of G, where du is the degree of a vertex u in G. In this paper, we show that HG≥DG+5/3−n/2 and HG≥1/2+2/3n−2DG, where G is a quasi-tree graph of order n and diameter DG. Indeed, we show that both lower bounds are tight and identify all quasi-tree graphs reaching these two lower bounds