Subgroup Graphs of Finite Groups

2020

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.

International Journal of Group Theory, 2021

‎This paper concerns aspects of various graphs whose vertex set is a group $G$‎ ‎and whose edges reflect group structure in some way (so that‎, ‎in particular‎, ‎they are invariant under the action of the automorphism group of $G$)‎. ‎The‎ ‎particular graphs I will chiefly discuss are the power graph‎, ‎enhanced power‎ ‎graph‎, ‎deep commuting graph‎, ‎commuting graph‎, ‎and non-generating graph‎. ‎My main concern is not with properties of these graphs individually‎, ‎but ‎‎‎‎rather with comparisons between them‎. ‎The graphs mentioned‎, ‎together‎ ‎with the null and complete graphs‎, ‎form a hierarchy (as long as $G$ is‎ ‎non-abelian)‎, ‎in the sense that the edge set of any one is contained in that‎ ‎of the next; interesting questions involve when two graphs in the hierarchy‎ ‎are equal‎, ‎or what properties the difference between them has‎. ‎I also ‎‎‎consider various properties such as universality and forbidden subgraphs‎, ‎comparing how these properties play out in the differ...

2021

The co-maximal subgroup graph Γ(G) of a group G is a graph whose vertices are nontrivial proper subgroups of G and two vertices H and K are adjacent if HK = G. In this paper, we continue the study of Γ(G), especially when Γ(G) has isolated vertices. We define a new graph Γ∗(G), which is obtained by removing isolated vertices from Γ(G). We characterize when Γ∗(G) is connected, a complete graph, star graph, has an universal vertex etc. We also find various graph parameters like diameter, girth, bipartiteness etc. in terms of properties of G.

Italian journal of pure and applied mathematics, 2016

The non-coprime graph ΠG,H of a finite groupG with respect to a subgroup H is a simple graph with G \ {e} as the vertex set. Two vertices u and v of this graph are adjacent if and only if (|u|, |v|) ̸= 1 and (u ∈ H or v ∈ H). In this paper, the main properties of this graph are obtained. The isomorphism problem of this graph is also investigated for nilpotent groups.

Journal of Algebra and Its Applications, 2014

The power graph [Formula: see text] of a group G is a simple graph whose vertex-set is G and two vertices x and y in G are adjacent if and only if one of them is a power of the other. The subgraph [Formula: see text] of [Formula: see text] is obtained by deleting the vertex 1 (the identity element of G). In this paper, we first investigate some properties of the power graph [Formula: see text] and its subgraph [Formula: see text]. We next provide necessary and sufficient conditions for a power graph [Formula: see text] to be a strongly regular graph, a bipartite graph or a planar graph. Finally, we obtain some infinite families of finite groups G for which the power graph [Formula: see text] contains some cut-edges.

ScienceAsia, 2015

We classify planar graphs and complete power graphs of groups and show that the only infinite group with a complete power graph is the Prüfer group p ∞. Clique and chromatic numbers and the automorphism group of power graphs are investigated. We also prove that the reduced power graph of a group G is regular if and only if G is a cyclic p-group or exp(G) = p for some prime number p.

2018

The power graph of an arbitrary group G is a simple graph with all elements of G as its vertices and two vertices are adjacent if one is a positive power of another. In this paper, we generalize this concept to a graph whose vertices are all elements of G that generate a proper subgroup of G and two elements are adjacent if the cyclic subgroup generated by which have non-trivial intersections. We concentrate on completeness and planarity of this graph.

2021

Let G be a finite group. A number of graphs with the vertex set G have been studied, including the power graph, enhanced power graph, and commuting graph. These graphs form a hierarchy under the inclusion of edge sets, and it is useful to study them together. In addition, several authors have considered modifying the definition of these graphs by choosing a natural equivalence relation on the group such as equality, conjugacy, or equal orders, and joining two elements if there are elements in their equivalence class that are adjacent in the original graph. In this way, we enlarge the hierarchy into a second dimension. Using the three graph types and three equivalence relations mentioned gives nine graphs, of which in general only two coincide; we find conditions on the group for some other pairs to be equal. These often define interesting classes of groups, such as EPPO groups, 2-Engel groups, and Dedekind groups. We study some properties of graphs in this new hierarchy. In particul...

Mathematics

Let H be a subgroup of a finite non-abelian group G and g∈G. Let Z(H,G)={x∈H:xy=yx,∀y∈G}. We introduce the graph ΔH,Gg whose vertex set is G\Z(H,G) and two distinct vertices x and y are adjacent if x∈H or y∈H and [x,y]≠g,g−1, where [x,y]=x−1y−1xy. In this paper, we determine whether ΔH,Gg is a tree among other results. We also discuss about its diameter and connectivity with special attention to the dihedral groups.

2012

Abstract. The power graph P (G) of a group G is the graph whose vertex set is the group elements and two elements are adjacent if one is a power of the other. In this paper, we consider some graph theoretical properties of a power graph P (G) that can be related to its group theoretical properties. As consequences of our results, simple proofs for some earlier results are presented.