Power Graph Research Articles

Characterizations of p-groups whose power graphs satisfy certain connectivity conditions

Let [Formula: see text] be an undirected and simple graph. A set [Formula: see text] of vertices in [Formula: see text] is called a cyclic vertex cutset of [Formula: see text] if [Formula: see text] is disconnected and has at least two components containing cycles. If [Formula: see text] has a cyclic vertex cutset, then it is said to be cyclically separable. The cyclic vertex connectivity of [Formula: see text] is the minimum of cardinalities of the cyclic vertex cutsets of [Formula: see text]. The power graph [Formula: see text] of a group [Formula: see text] is the undirected and simple graph whose vertices are the elements [Formula: see text] and two vertices are adjacent if one of them is the power of other in [Formula: see text]. In this paper, we first characterize the finite [Formula: see text]-groups ([Formula: see text] is a prime number) whose power graphs are cyclically separable in terms of their maximal cyclic subgroups. Then, we characterize the finite [Formula: see text]-groups whose power graphs have equal vertex connectivity and cyclic vertex connectivity.

Forbidden Subgraphs in Intersection Power Graphs of Finite Groups

Forbidden Subgraphs in Intersection Power Graphs of Finite Groups

On Spectrum and Energies of Enhanced Power Graphs

On Spectrum and Energies of Enhanced Power Graphs

On the self-complementary power graph of finite groups

On the self-complementary power graph of finite groups

Power graphs and a combinatorial property of abundant semigroups

The aim of this paper is to study the power graph of a semigroup. We obtain sufficient and necessary conditions for the independence number of the power graph of an (IC) abundant semigroups (superabundant semigroups) to be finite.

Planarity of the bipartite graph associated to squares of elements and subgroups of a group

A new kind of bipartite graph is defined in this paper. The work is motivated mainly from a graph introduced by Al-Kaseasbeh and Erfanian [A bipartite graph associated to elements and cosets of subgroups of a finite group, AIMS Math. 6(10) (2021) 10395–10404] and partially from the undirected power graph of a group and from the inclusion graph. For a nontrivial group [Formula: see text], we define a simple undirected bipartite graph [Formula: see text] with the vertex set [Formula: see text], where [Formula: see text] is the set of all elements of the group [Formula: see text] and [Formula: see text] is the set of all subgroups [Formula: see text] of [Formula: see text] such that [Formula: see text] and two vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. Here, we classify all the finite groups [Formula: see text] whose bipartite graphs [Formula: see text] are planar. In addition, we also classify the finite groups [Formula: see text] such that [Formula: see text] is outerplanar, maximal planar, maximal outerplanar, star, tree, claw graph, respectively. The planarity of the graph [Formula: see text] corresponding to an infinite group [Formula: see text] has also been investigated here. It is also observed that [Formula: see text] satisfies Vizing’s conjecture.

Some properties of proper power graphs in finite Abelian groups

The power graph of a group [Formula: see text], denoted as [Formula: see text], constitutes a simple undirected graph characterized by its vertex set [Formula: see text]. Specifically, vertices [Formula: see text] exhibit adjacency exclusively if [Formula: see text] belongs to the cyclic subgroup generated by [Formula: see text] or vice versa. The corresponding proper power graph of [Formula: see text] is obtained by taking [Formula: see text] and removing a vertex corresponding to the identity element, which is denoted as [Formula: see text]. In the context of finite abelian groups, this paper establishes the sufficient and necessary conditions for the proper power graph’s connectedness. Moreover, a precise upper bound for the diameter of [Formula: see text] in finite abelian groups is provided with sharpness. This paper also explores the study of vertex connectivity, center, and planarity.

Drug-target binding affinity prediction based on power graph and word2vec

BackgroundDrug and protein targets affect the physiological functions and metabolic effects of the body through bonding reactions, and accurate prediction of drug-protein target interactions is crucial for drug development. In order to shorten the drug development cycle and reduce costs, machine learning methods are gradually playing an important role in the field of drug-target interactions.ResultsCompared with other methods, regression-based drug target affinity is more representative of the binding ability. Accurate prediction of drug target affinity can effectively reduce the time and cost of drug retargeting and new drug development. In this paper, a drug target affinity prediction model (WPGraphDTA) based on power graph and word2vec is proposed.ConclusionsIn this model, the drug molecular features in the power graph module are extracted by a graph neural network, and then the protein features are obtained by the Word2vec method. After feature fusion, they are input into the three full connection layers to obtain the drug target affinity prediction value. We conducted experiments on the Davis and Kiba datasets, and the experimental results showed that WPGraphDTA exhibited good prediction performance.

Drug target affinity prediction based on multi-scale gated power graph and multi-head linear attention mechanism.

For the purpose of developing new drugs and repositioning existing ones, accurate drug-target affinity (DTA) prediction is essential. While graph neural networks are frequently utilized for DTA prediction, it is difficult for existing single-scale graph neural networks to access the global structure of compounds. We propose a novel DTA prediction model in this study, MAPGraphDTA, which uses an approach based on a multi-head linear attention mechanism that aggregates global features based on the attention weights and a multi-scale gated power graph that captures multi-hop connectivity relationships of graph nodes. In order to accurately extract drug target features, we provide a gated skip-connection approach in multiscale graph neural networks, which is used to fuse multiscale features to produce a rich representation of feature information. We experimented on the Davis, Kiba, Metz, and DTC datasets, and we evaluated the proposed method against other relevant models. Based on all evaluation metrics, MAPGraphDTA outperforms the other models, according to the results of the experiment. We also performed cold-start experiments on the Davis dataset, which showed that our model has good prediction ability for unseen drugs, unseen proteins, and cases where neither drugs nor proteins has been seen.

Metric and strong metric dimension in TI-power graphs of finite groups

<p>Given a finite group $ G $ with the identity $ e $, the TI-power graph (trivial intersection power graph) of $ G $ is an undirected graph with vertex set $ G $, in which two distinct vertices $ x $ and $ y $ are adjacent if $ \langle x\rangle\cap \langle y\rangle = \{e\} $. In this paper, we obtain closed formulas for the metric and strong metric dimensions of the TI-power graph of a finite group. As applications, we compute the metric and strong metric dimensions of the TI-power graph of a cyclic group, a dihedral group, a generalized quaternion group, and a semi-dihedral group.</p>

Critical classes of power graphs and reconstruction of directed power graphs

Critical classes of power graphs and reconstruction of directed power graphs

ON NEIGHBOURHOODS IN THE ENHANCED POWER GRAPH ASSOCIATED WITH A FINITE GROUP

Abstract We investigate neighbourhood sizes in the enhanced power graph (also known as the cyclic graph) associated with a finite group. In particular, we characterise finite p-groups with the smallest maximum size for neighbourhoods of a nontrivial element in its enhanced power graph.

Adjacency Spectrum of Reduced Power Graphs of Finite Groups

Adjacency Spectrum of Reduced Power Graphs of Finite Groups

On the power graphs of semigroups of homogeneous elements of graded rings

On the power graphs of semigroups of homogeneous elements of graded rings

Metric and strong metric dimension in intersection power graphs of finite groups

Let G be a finite group and let e be its identity element. The intersection power graph of G is the undirected graph with vertex set G, in which two distinct vertices x and y are adjacent if either 〈 x 〉 ∩ 〈 y 〉 ≠ { e } or one of x and y is e. In this paper, we first compute the strong metric dimension of the intersection power graph of a cyclic group, a dihedral group, and a generalized quaternion group. We also characterize all finite groups G whose intersection power graph has strong metric dimension | G | − 2 . Then, we give the sharp upper and lower bounds for the metric dimension of the intersection power graph of a finite group. As applications, we obtain the metric dimension of the intersection power graph of a cyclic group, a dihedral group, a generalized quaternion group, and a group of odd order.

On the strong domination number of proper enhanced power graphs of finite groups

On the strong domination number of proper enhanced power graphs of finite groups

Automorphism group of the reduced power (di)graph of a finite group

We describe the full automorphism group of the directed reduced power graph and the undirected reduced power graph of a finite group. We compute the full automorphism groups of these graphs of several classes of finite groups. Also, we establish some relation between the automorphism group of the undirected reduced power graph (resp. the directed reduced power graph) and the automorphism group of the undirected power graph (resp. the directed power graph) of a finite group.

Genus and crosscap of normal subgroup based power graphs of finite groups

Genus and crosscap of normal subgroup based power graphs of finite groups

Notes on the diameter of the complement of the power graph of a finite group

Notes on the diameter of the complement of the power graph of a finite group

Algebraic Connectivity of Power Graphs of Finite Cyclic Groups

Algebraic Connectivity of Power Graphs of Finite Cyclic Groups