Abstract The prime graph , also called the Gruenberg–Kegel graph , of a finite group 𝐺 is the labelled graph Γ ⁢ ( G ) \Gamma(G) with vertices the prime divisors of | G | \lvert G\rvert and edges the pairs { p , q } \{p,q\} for which 𝐺 contains an element of order p ⁢ q pq . A group 𝐺 is recognisable by its prime graph if every group 𝐻 with Γ ⁢ ( H ) = Γ ⁢ ( G ) \Gamma(H)=\Gamma(G) is isomorphic to 𝐺. Cameron and Maslova have shown that every group that is recognisable by its prime graph is almost simple. This justifies the significant amount of attention that has been given to determining which simple or almost simple groups are recognisable by their prime graphs. This problem has been completely solved for certain families of simple groups, including the sporadic groups. A natural extension of the problem is to determine which groups are recognisable by their unlabelled prime graphs, i.e. by the isomorphism types of their prime graphs. Here we determine which of the sporadic finite simple groups are recognisable by the isomorphism types of their prime graphs. We also show that, for every sporadic group 𝐺 that is not recognisable by the isomorphism type of Γ ⁢ ( G ) \Gamma(G) , there are infinitely many groups 𝐻 with Γ ⁢ ( H ) ≅ Γ ⁢ ( G ) \Gamma(H)\cong\Gamma(G) .

Abstract We study the finite solvable groups 𝐺 in which every real element has prime power order. We divide our examination into two parts: the case O 2 ⁢ ( G ) > 1 \mathbf{O}_{2}(G)>1 and the case O 2 ⁢ ( G ) = 1 \mathbf{O}_{2}(G)=1 . Specifically we prove that if O 2 ⁢ ( G ) > 1 \mathbf{O}_{2}(G)>1 , then 𝐺 is a { 2 , p } \{2,p\} -group. Finally, by taking into consideration the examples presented in the analysis of the O 2 ⁢ ( G ) = 1 \mathbf{O}_{2}(G)=1 case, we deduce some interesting and unexpected results about the connectedness of the real prime graph Γ R ⁢ ( G ) \Gamma_{\mathbb{R}}(G) . In particular, we find that there are groups such that Γ R ⁢ ( G ) \Gamma_{\mathbb{R}}(G) has 3 or 4 connected components.

Classifying prime graphs of finite groups – a methodical approach

A graph G is considered to have a prime labeling when each of its n vertices is assigned a unique label from the set {1, 2, 3, 4, ..., n}, ensuring that the labels of any two connected vertices are coprime. In the literature, many graph classes identified as prime graphs and non-prime graphs. In this paper, we focus on converting non-prime graph classes to a prime graph classes by applying corona product with complete graph on one vertex.

Graph labeling is the assigning of labels represented by integers or symbols to graph elements, edges and/or vertices (or both) of a graph. Consider a simple graph with a vertex-set and an edge-set . The order of graph , denoted by , is the number of vertices on . The prime labeling is a bijective function , such that the labels of any two adjacent vertices in G are relatively prime or , for every two adjacent vertices and in . If a graph can be labeled with prime labeling, then the graph can be said to be a prime graph. A flower graph is a graph formed by helm graph by connecting its pendant vertices (the vertices have degree one) to the central vertex of , such a flower graph is denoted as In this research, we employ constructive and analytical methods to investigate prime labelings on specific graph classes. Definitions, lemmas, and theorems are developed as the main results in this research. The amalgamation is a graph formed by taking all by taking all the and identifying their fixed vertices . If , then we write with . In previous research, it has been shown that the flower graphs , for are prime graphs. Continuing the research, we prove that two classes of amalgamation of flower graphs are prime graphs.

Abstract We obtain restrictions on units of even order in the integral group ring of a finite group by studying their actions on the reductions modulo 4 of lattices over the 2‐adic group ring . This improves the “lattice method” which considers reductions modulo primes , but is of limited use for essentially due to the fact that . Our methods yield results in cases where has blocks, whose defect groups are Klein four groups or dihedral groups of order 8. This allows us to disprove the existence of units of order for almost simple groups with socle where and to answer the prime graph question affirmatively for many such groups.

In this research work, we study the commutativity in ring R in relation with strong commutativity preserving maps (SCP). We extend the applications of our results in R to prime graphs. Let P be a nonempty subset of a ring R. A mapping ????: ???? → ???? is said to be strong commutativity preserving maps on P if [????(????), ????(????)] = [????, ????] for all ????, ???? ∈ ????.

Graph theory has become a key tool in group theory research, enabling the investigation of algebraic structures through graph properties. This has led to the evolution of diverse group graph definitions. This study explored the connection between alternating groups and their order product prime graphs. We constructed the order product prime graphs for alternating groups of degree 5 and 6, analyzing properties like regularity, completeness, connectivity, girth and diameter. Using 'Group Algorithm Programming Software' we generated group elements and built the graphs. Our findings revealed that the order product prime graphs of alternating groups of degrees 5 and 6 are connected, irregular, incomplete, and non-planar, with a consistent topology featuring a girth of 3 and diameter of 2, indicating a stable and well-connected structure.

An odd prime labeling of a graph G (V,E), is defined as a bijective function f mapping the vertex set V to the set {1,3,5,…,2|V(G)|1}, such that for every edge uvE. the greatest common divisor gcd(f(u),f(v))=1. A graph that permits such a labeling is referred to as an odd prime graph. In this study, we explore the odd prime labeling properties of various graph structures, including the octopus chain graph, octopus ladder graph, twisted octopus ladder graph, and hexa-octopus chain graph.

Finite Solvable Groups Whose Prime Graphs have Diameter 3

Graph-based approaches have emerged as a key area of research in group theory, with a growing focus on leveraging graph properties to investigate algebraic structures. Research on the graphical structure of finite groups has led to the development of various definitions of group graphs over time. This study explored the relationship between alternating groups and their respective order product prime graphs. The research constructed the order product prime graphs for alternating groups of degree three and four. Regularity, completeness, connectivity, girth and diameter of each graph constructed were also checked. The study utilized the 'Group Algorithm Programming Software' to generate the elements of the alternating groups investigated, facilitating the construction of their order product prime graphs. The findings revealed that the order product prime graphs of alternating groups of degrees three and four are connected. The study discovered that the graph of degree three has the distinctive properties of completeness, regularity, and planarity, a trait that sets it apart from the graphs of degree four. The graph of degree three has a girth of 3 and a diameter of 1, indicating a highly connected and compact structure. Notably, the graphs of degree four, exhibits a girths of 3 and diameters of 2, indicating a stable and well-connected structure.

The category of directed graphs is isomorphic to a particular category whose objects are labeled undirected bipartite graphs and whose morphisms are undirected graph morphisms that respect the labeling. Based on this isomorphism, we begin by showing that the class of all directed graphs is a Graph Isomorphism Complete class. Afterwards, by extending this categorical framework to weighted prime graphs, we prove that the categories of multidirected graphs with and without self-loops are each isomorphic to a particular category of weighted prime graphs. Consequently, we prove that these classes of multidirected graphs are also Graph Isomorphism Complete.

The prime graph P G ( R ) of a ring R is a graph whose vertex set consists of all elements of R . Two elements x , y ∈ R are adjacent in the graph if and only if x R y = 0 or y R x = 0. An element a ∈ R is called a strong zero divisor in R if 〈 a 〉〈 b 〉 = 0 or 〈 b 〉〈 a 〉 = 0 for some nonzero element b ∈ R . The set of all strong zero divisors is denoted by S ( R ). In this paper, we study the prime graph of a ring R , considering S ( R ) as the set of vertices. In this way, we introduce the modified prime graph P G ∗ ( R ). We then investigate some combinatorial properties of the prime graphs P G ( R [ x ]) and P G ( R [[ x ]]) such as completeness, diameter, and girth, where R is a noncommutative ring.

The chromatic numbers of prime graphs of polynomials and power series over rings

This paper investigates a novel characterization method for the sporadic simple group $B$ (Baby Monster Group). Using the combination of the order component set $OC(G)$ of a finite group and the set $\pi_{p_m}(G)$ of the orders of centralizers of elements of highest order within the group, we proved: a finite group $G$ is isomorphic to the sporadic simple group $B$ if and only if the following two conditions hold: (1) $G$ shares the same largest order component $m_1(G)$ as B; (2) The sets $\pi_{p_m}(G)$ and $\pi_{p_m}(B)$ of the orders of centralizers of their respective highest-order elements are identical. By analyzing the prime graph structure, excluding the possibility of Frobenius groups and 2-Frobenius groups, and utilizing the Classification Theorem of Finite Simple Groups, we ultimately establish the sufficiency and necessity of these characterizing conditions.

A graph G = (V (G), E(G)) is observed to admit prime labeling, if a graph that receives prime labeling is called prime graph .In this research article we investigate that the Goldner Harary graph admits prime labeling. We establish prime labeling using some graph operations such as duplication. Switching and fusion with few ideas.

Abstract In the context of a simple undirected graph G G , a k k -prime labeling refers to assigning distinct integers from the set { k , k + 1 , … , ∣ V ( G ) ∣ + k − 1 } \left\{k,k+1,\ldots ,| V\left(G)| +k-1\right\} to its vertices, such that adjacent vertices in G G are labeled with numbers that are relatively prime to each other. If G G has a k k -prime labeling, we say that G G is a k k -prime graph (k-PG). In this article, we characterize when a graph up to order 6 is a k-PG and characterize when a graph of order 7 is a k-PG whenever k k and k + 1 k+1 are not divisible by 5. Also, we find a lower bound for the independence number of a k-PG. Finally, we study when a cycle is a k-PG.

Let G be a graph. A bijection f:V→ {1,2,…..|V|} is called a prime labeling [3] if for each edge e=uv in E, we have GCD{ f(u),f(v)}=1. A graph that admits a prime labeling is said to be a prime graph. In this paper we show that bull graph admits Prime labeling in the context of variety graph operations namely duplication of vertex, fusion of vertices and Switching in Bull graph.

A prime labeling of a graph G is a map from the vertex set of G to the set {1, 2, ..., |V (G)|} such that any two adjacent vertices in the graph G have labels that are relatively prime. In this paper, we discuss when the disjoint union of some graphs is a prime graph.

The Prime Coprime Graph is defined as a graph in which two distinct vertices are adjacent if and only if the greatest common divisor of their orders is 1, indicating that they are coprime. This research focuses on deriving general formulas for the Padmakar-Ivan index and the Szeged index for the coprime prime graph of the modulo integer group with n=p^k, where p is a prime number and k is not less than 2. As a result of this study, explicit formulas for the Padmakar-Ivan and Szeged indices were obtained, along with an analysis of the relationship between these two indices.Keywords: prime coprime graph, Padmakar-Ivan index, Szeged index.