Graphs of nonsolvable groups with four degree-vertices

Let G be a finite group. The degree(vertex) graph Γ(G) attached to G is a character degree graph.Its vertices are the degrees of the nonlinear irreducible complex characters of G, and different vertices m, n are adjacent if the greatest common divisor(m, n) 1. In this paper, we classify all graphs with four vertices that occur as Γ(G) for nonsolvable groups G.     

On a Commuting Graph on Conjugacy Classes of Groups

We consider the graph Γ(G), associated with the conjugacy classes of a group G. Its vertices are the nontrivial conjugacy classes of G, and we join two different classes C, D, whenever there exist x ∈ G and y ∈ D such that xy = yx. The aim of this article is twofold. First, we investigate which graphs can occur in various contexts and second, given a graph Γ(G) associated with G, we investigate the possible structure of G. We proved that if G is a periodic solvable group, then Γ(G) has at most two components, each of diameter at most 9. If G is any locally finite group, then Γ(G) has at most 6 components, each of diameter at most 19. Finally, we investigated periodic groups G with Γ(G) satisfying one of the following properties: (i) no edges exist between noncentral conjugacy classes, and (ii) no edges exist between infinite conjugacy classes. In particular, we showed that the only nonabelian groups satisfying (i) are the three finite groups of order 6 and 8.     

Commuting Graphs of Matrix Algebras

The commuting graph of a ring R, denoted by Γ(R), is a graph whose vertices are all noncentral elements of R, and two distinct vertices x and y are adjacent if and only if xy = yx. The commuting graph of a group G, denoted by Γ(G), is similarly defined. In this article we investigate some graph-theoretic properties of Γ(M _n (F)), where F is a field and n ≥ 2. Also we study the commuting graphs of some classical groups such as GL _n (F) and SL _n (F). We show that Γ(M _n (F)) is a connected graph if and only if every field extension of F of degree n contains a proper intermediate field. We prove that apart from finitely many fields, a similar result is true for Γ(GL _n (F)) and Γ(SL _n (F)). Also we show that for two fields F and E and integers n, m ≥ 2, if Γ(M _n (F))?Γ(M _m (E)), then n = m and |F|=|E|.     

Commuting Graphs of Group Algebras

The commuting graph of a ring R, denoted by Γ(R), is a graph of all whose vertices are noncentral elements of R, and 2 distinct vertices x and y are adjacent if and only if xy = yx. In this article we investigate some graph-theoretic properties of Γ(kG), where G is a finite group, k is a field, and 0 ≠ |G| ∈k. Among other results it is shown that if G is a finite nonabelian group and k is an algebraically closed field, then Γ(kG) is not connected if and only if |G| = 6 or 8. For an arbitrary field k, we prove that Γ(kG) is connected if G is a nonabelian finite simple group or G′ ≠ G″ and G″ ≠ 1.     

Commutativity Pattern of Finite Non-Abelian p-Groups Determine Their Orders

Let G be a non-abelian group and Z(G) be the center of G. Associate a graph Γ _G (called noncommuting graph of G) with G as follows: Take G?Z(G) as the vertices of Γ _G , and join two distinct vertices x and y, whenever xy ≠ yx. Here, we prove that “the commutativity pattern of a finite non-abelian p-group determine its order among the class of groups"; this means that if P is a finite non-abelian p-group such that Γ _P  ? Γ _H for some group H, then |P| = |H|.     

Noncyclic Graph of a Group

We associate a graph Γ _G to a nonlocally cyclic group G (called the noncyclic graph of G) as follows: take G\ Cyc(G) as vertex set, where Cyc(G) = {x ? G| 〈x, y〉 is cyclic for all y ? G}, and join two vertices if they do not generate a cyclic subgroup. We study the properties of this graph and we establish some graph theoretical properties (such as regularity) of this graph in terms of the group ones. We prove that the clique number of Γ _G is finite if and only if Γ _G has no infinite clique. We prove that if G is a finite nilpotent group and H is a group with Γ _G  ? Γ _H and |Cyc(G)| = |Cyc(H)| = 1, then H is a finite nilpotent group. We give some examples of groups G whose noncyclic graphs are “unique”, i.e., if Γ _G  ? Γ _H for some group H, then G ? H. In view of these examples, we conjecture that every finite nonabelian simple group has a unique noncyclic graph. Also we give some examples of finite noncyclic groups G with the property that if Γ _G  ? Γ _H for some group H, then |G| = |H|. These suggest the question whether the latter property holds for all finite noncyclic groups.     

The Orders of Vertices of a Character-degree Graph

Let G be a finite group. We put ρ(G) = {p|p is a prime dividing χ(1) for some χ ∈Irr(G)}. We define a graph Γ(G), whose vertices are the primes in ρ(G) and p, q ∈ ρ(G) are connected in Γ(G) denoted p ～ q, if pq||χ(1) for some χ ∈Irr(G). For p ∈ ρ(G), we define ord(p) = |{q ∈ ρ(G)|q ～ p}|. We call ord(p) the order of the vertex p of the graph Γ(G). In this article, we discuss orders and the influences of orders on the structure of finite groups.     

Finite Groups with a Disconnected p -Regular Conjugacy Class Graph

Abstract

Let G be a finite p-solvable group for a fixed prime p. Attach to G a graph Γ _p (G) whose vertices are the non-central p-regular conjugacy classes of G and connect two vertices by an edge if their cardinalities have a common prime divisor. In this note we study the structure and arithmetical properties of the p-regular class sizes in p-solvable groups G having Γ _p (G) disconnected.     

On Commuting Graphs of Finite Matrix Rings

The commuting graph of a ring R, denoted by Γ(R), is a graph whose vertices are all noncentral elements of R and two distinct vertices are joint by an edge whenever they commute. It is conjectured that if R is a ring with identity such that Γ(R) ≈ Γ(M _n (F)), for a finite field F and n ≥ 2, then R ≈ M _n (F). Here we prove this conjecture when n = 2.     

The Zero Divisor Graph of 2 × 2 Matrices Over a Field

A zero divisor graph, Γ(R), is formed from a ring R by having each element of Z(R) \ {0} to be a vertex in the graph and having two vertices u and v adjacent if the corresponding elements from the ring are nonequal and have product equal to zero. In this paper, the structure of the zero-divisor graph of 2 × 2 matrices over a field, Γ(M2(F)), are completely determined.     

Finite Groups with Four Conjugacy Class Sizes

We determine the structure of all finite groups with four class sizes when two of them are coprime numbers larger than 1. We prove that such groups are solvable and that the set of class sizes is exactly {1, m, n, mk}, where m, n > 1 are coprime numbers and k > 1 is a divisor of n.     

An answer to a question of Isaacs on character degree graphs

Let N be a normal subgroup of a finite group G. We consider the graph Γ(G|N) whose vertices are the prime divisors of the degrees of the irreducible characters of G whose kernel does not contain N and two vertices are joined by an edge if the product of the two primes divides the degree of some of the characters of G whose kernel does not contain N. We prove that if Γ(G|N) is disconnected then G/N is solvable. This proves a strong form of a conjecture of Isaacs.     

Dominating Sets of Some Graphs Associated to Commutative Rings

Let G = (V, E) be a graph. A set S ? V is a dominating set of G if every vertex in V is either in S or is adjacent to a vertex in S. The domination number γ(G) of G is the minimum cardinality among the dominating sets of G. The main object of this article is to study and characterize the dominating sets of the zero-divisor graph Γ(R) and ideal-based zero-divisor graph Γ _I (R) of a commutative ring R.     

Some Remarks on the Graph Γ I (R)

Let R be a commutative ring with nonzero identity and Z(R) its set of zero-divisors. The zero-divisor graph of R is Γ(R), with vertices Z(R)?{0} and distinct vertices x and y are adjacent if and only if xy = 0. For a proper ideal I of R, the ideal-based zero-divisor graph of R is Γ _I (R), with vertices {x ∈ R?I | xy ∈ I for some y ∈ R?I} and distinct vertices x and y are adjacent if and only if xy ∈ I. In this article, we study the relationship between the two graphs Γ(R) and Γ _I (R). We also determine when Γ _I (R) is either a complete graph or a complete bipartite graph and investigate when Γ _I (R) ? Γ(S) for some commutative ring S.     

The generating graph of finite soluble groups

For a finite group G, let Γ(G) denote the graph defined on the non-identity elements of G in such a way that two distinct vertices are connected by an edge if and only if they generate G. We prove that if G is soluble, then the non-isolated vertices of Γ(G) belong to a unique connected component.     

Non-Commuting Graphs of Nilpotent Groups

Let G be a non-abelian group and Z(G) be the center of G. The non-commuting graph Γ _G associated to G is the graph whose vertex set is G?Z(G) and two distinct elements x, y are adjacent if and only if xy ≠ yx. We prove that if G and H are non-abelian nilpotent groups with irregular isomorphic non-commuting graphs, then |G| = |H|.     

On Groups with Given Same-Order Types

On any group G, define g ～ h if g, h ∈ G have the same order. The set of sizes of the equivalence classes with respect to this relation is called the same-order type of G. In this article we prove that a group of the same-order type {1, n} is nilpotent and of the same-order type {1, m, n} is solvable.     

On stress matrices of (d + 1)-lateration frameworks in general position

Let (G, P) be a bar framework of n vertices in general position in ${\mathbb{R}^d}$ , for d ≤ n ? 1, where G is a (d + 1)-lateration graph. In this paper, we present a constructive proof that (G, P) admits a positive semidefinite stress matrix with rank (n ? d ? 1). We also prove a similar result for a sensor network, where the graph consists of m(≥ d + 1) anchors.     

Graphs with given group and given constant link

A graph L is called a link graph if there is a graph G such that for each vertex of G its neighbors induce a subgraph isomorphic to L. Such a G is said to have constant link .L Sabidussi proved that for any finite group F and any n ? 3 there are infinitely many n-regular connected graphs G with AutG ? Γ. We will prove a stronger result: For any finite group Γ and any link graph L with at least one isolated vertex and at least three vertices there are infinitely many connected graphs G with constant link L and AutG ? Γ.     

Lollipop graphs are extremal for commute times

Consider a simple random walk on a connected graph G=(V, E). Let C(u, v) be the expected time taken for the walk starting at vertex u to reach vertex v and then go back to u again, i.e., the commute time for u and v, and let C(G)=max_{u, v∈V}C(u, v). Further, let 𝒢(n, m) be the family of connected graphs on n vertices with m edges, , and let 𝒢(n)=∪_m𝒢(n, m) be the family of all connected n‐vertex graphs. It is proved that if G∈(n, m) is such that C(G)=max_{H∈𝒢(n, m)}C(H) then G is either a lollipop graph or a so‐called double‐handled lollipop graph. It is further shown, using this result, that if C(G)=max_H∈𝒢(n)C(H) then G is the full lollipop graph or a full double‐handled lollipop graph with [(2n−1)/3] vertices in the clique unless n≤9 in which case G is the n‐path. ©2000 John Wiley & Sons, Inc. Random Struct. Alg., 16, 131–142, 2000