English

<p style="text-align: left;" dir="ltr"> Let <em>G</em> be a finite group and let <em>N</em> be a fixed normal subgroup of <em>G</em>. In this paper, a new kind of graph on <em>G</em>, namely the intersection graph is defined and studied. We use <img src="/public/site/images/ikhsan/equation.png" alt="" width="6" height="4" /> to denote this graph, with its vertices are all normal subgroups of <em>G</em> and two distinct vertices are adjacent if their intersection in <em>N</em>. We show some properties of this graph. For instance, the intersection graph is a simple connected with diameter at most two. Furthermore we give the graph structure of <img src="/public/site/images/ikhsan/equation_(1).png" alt="" width="6" height="4" /> for some finite groups such as the symmetric, dihedral, special linear group, quaternion and cyclic groups. </p>

Let G be a group. Recall that the intersection graph of subgroups of G is an undirected graph whose vertex set is the set of all nontrivial subgroups of G and distinct vertices H,K are joined by an edge in this graph if and only if the intersection of H and K is nontrivial. The aim of this article is to investigate the interplay between the group-theoretic properties of a finite group G and the graph-theoretic properties of the complement of the intersection graph of subgroups of G.

The nilpotent graph of a group G is a simple graph whose vertex set is G∖nil(G), where nil(G) = {y ∈ G | ⟨ x, y ⟩ is nilpotent ∀ x ∈ G}, and two distinct vertices x and y are adjacent if ⟨ x, y ⟩ is nilpotent. In this article, we show that the collection of finite non-nilpotent groups whose nilpotent graphs have the same genus is finite, derive explicit formulas for the genus of the nilpotent graphs of some well-known classes of finite non-nilpotent groups, and determine all finite non-nilpotent groups whose nilpotent graphs are planar or toroidal.

Given a finite group G, the co-prime order graph of G is the simple undirected graph whose vertex set is G, and two distinct vertices x; y are adjacent if gcd(o(x); o(y)) = 1 or p, where p is a prime, and o(x) and o(y) are the orders of x and y, respectively. In this paper, we prove that, for a fixed positive integer k, there are finitely many finite groups whose co-prime order graphs have (non)orientable genus k. As applications, we classify all finite groups whose co-prime order graphs have (non)orientable genus one and two.

In recent years, the graph used as a representation of a finite group. One kind of graph that represents a finite group is the non-coprime graph. The non-coprime graph of a finite group is simple graph where vertices are all elements of that group without identity element and two distinct vertices are adjacent if and only if its order is not coprime. In this research, we will discuss the non-coprime graph of a generalized quaternion group and its properties. The method that is used is to study literature and analyze it by finding patterns in various examples. The results of this research are the form of the graph, degree of each vertex, radius, diameter, girth, and total of cycles contained in the graph when n = 2k and n an odd prime number.

In this paper, a new kind of graph on a finite group , namely the result involution graph is defined and studied. We use to denote this graph, is a simple undirected graph with vertex set. Two distinct vertices are adjacent if and only if their product is nontrivial involution element in . The result involution graph for several finite groups are obtained. We study some properties of the result involution graph by resizing graph by using the conjugacy classes of . Finally, we show that the result involution graphs for the symmetric groups and the alternating groups are connected with diameter at most 3 and radius at most 2 for. Furthermore, they have girth 3.

Let G be a finite group. The prime graph Γ(G) of G is defined as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p, p′ are joined by an edge if G has an element of order pp′. Let L=Ln(2) or Un(2), where n≧17. We prove that L is quasirecognizable by prime graph, i.e. if G is a finite group such that Γ(G)=Γ(L), then G has a unique nonabelian composition factor isomorphic to L. As a consequence of our result we give a new proof for the recognition by element orders of Ln(2). Also we conclude that the simple group Un(2) is quasirecognizable by element orders.

Let G be a group. In this paper, we introduce and study the intersection subgroup graph of G as the (simple) graph with vertices the nontrivial subgroups H of G such that for some nontrivial subgroup K of G, and distinct vertices H and K are adjacent if and only if . We investigate when is a nonempty, finite, complete, complete bipartite, or planar graph.

Let R be a commutative ring, the Pseudo – Von Neumann regular graph of the ring R is define as a graph whose vertex set consists of all elements of R and any two distinct vertices a and b are adjacent if and only if , this graph is denoted by P-VG(R), in this work we got some new results about chromatic number of Pseudo-Von Neumann regular graph of cartesian product of rings.

The co-maximal subgroup graph Γ(G), of a finite group G, is a special kind of undirected graph, in which the set of vertices contains non-trivial proper subgroups of G and two distinct vertices A and U are adjacent iff AU = G. In this paper, we discuss the structures of co-maximal subgroup graphs of finite cyclic groups ℤpn, ℤp1np2m, ℤp1np2p3, ℤpn1pm2p3 and ℤp1np2mpl3 for n, m, l ∈ N, where p1, p2 and p3 are distinct prime numbers. Also, we establish the super graceful labeling of these graphs.

An edge-coloured graph G is said to be rainbow-connected if any two vertices are connected by a path whose edges have different colours. The rainbow connection number of a graph is the minimum number of colours needed to make the graph rainbow-connected. This parameter was introduced by G. Chartrand, G.L. Johns, K.A. McKeon and P. Zhang in 2008. Similar to rainbow connection coloring, an edge-coloring is a rainbow k-connection coloring if there are at least k internally disjoint rainbow u – v paths connecting any two distinct vertices u and v. And the rainbow k-connectivity rck (G) of G to be the minimum integer l such that there exists a l-edge-coloring which is a rainbow k-connection coloring. In [4], the authors determined the rck (G) of the complete graph Kn and r-regular complete bipartite graphs Kr, r . In this paper, we study the rainbow k-connectivity of the Cartesian product graphs K 2□Kn for some k. We determine rck (K 2□Kn ) for k = 2, 3, 4 and prove that there exists Cartesian product graphs K 2□Kn such that rck (K 2□Kn ) = 3 for each integer k ≥ 2.

The power graph of a group G is the simple graph with vertex set G and two distinct vertices are adjacent if one of them is a positive power of the other. For a finite noncyclic nilpotent group G, we study the minimum degree of Under some conditions involving the prime divisors of and the Sylow subgroups of G, we identify certain vertices associated with the generators of maximal cyclic subgroups of G such that is the degree of one of them. As an application, we obtain for some classes of nilpotent groups G.

Let G be a finite abelian group with identity 0. For an integer the additive power graph of G is the simple undirected graph with vertex set G in which two distinct vertices x and y are adjacent if and only if x + y = nt for some with When the additive power graph has been studied in the name of square graph of finite abelian groups. In this paper, we study the additive power graph of G with n = 3 and name the graph as the cubic power graph. The cubic power graph of G is denoted by More specifically, we obtain the diameter and the girth of the graph and its complement Using these, we obtain a condition for and its complement to be self-centered. Also, we obtain the independence number, the clique number and the chromatic number of and its complement and hence we prove that and its complement are weakly perfect. Also, we discuss about the perfectness of At last, we obtain a condition for and its complement to be vertex pancyclic.

On normality of n-Cayley graphs

Let G be a finite group, and let $$\textrm{cd}(G)$$ denote the set of degrees of the irreducible complex characters of G. Define then the character degree graph $$\Delta (G)$$ as the (simple undirected) graph whose vertices are the prime divisors of the numbers in $$\textrm{cd}(G)$$ , and two distinct vertices p, q are adjacent if and only if pq divides some number in $$\textrm{cd}(G)$$ . This paper continues the work, started in [8], toward the classification of the finite non-solvable groups whose degree graph possesses a cut-vertex, i.e. a vertex whose removal increases the number of connected components of the graph. While, in [8], groups with no composition factors isomorphic to $$\textrm{PSL}_{2}(t^a)$$ (for any prime power $$t^a\ge 4$$ ) were treated, here we consider the complementary situation in the case when $$t$$ is odd and $$t^a> 5$$ . The proof of this classification will be then completed in the third and last paper of this series [7] that deals with the case $$t=2$$ .