Power Graph Research Articles

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

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

Let S be a groupoid (magma) with zero 0, and let R = ⊕ s ∈ S R s be a contracted S-graded ring, that is, an S-graded ring with R 0 = 0. By G ( H R ) we denote the undirected power graph of a multiplicative subsemigroup H R = ∪ s ∈ S R s of R, and by G * ( H R ) a graph obtained from G ( H R ) by removing 0 and its incident edges. If Re is a nonzero ring component of R, then G * ( R e ) denotes a subgraph of G * ( H R ) , induced by R e * . In this paper we address a problem raised in [Abawajy, J., Kelarev, A., Chowdhury, M.: Power Graphs: A Survey. Electron. J. Graph Theory Appl. 1(2), 125–147 (2013)]. Namely, let S be torsion-free, that is, s n = t n implies s = t for all s, t ∈ S , and all positive integers n, and let S be 0-cancellative, that is, for all s, t, u ∈ S , su = tu ≠ 0 implies s = t , and us = ut ≠ 0 implies s = t . Also, let R be semisimple Artinian. We prove that if G * ( R e ) is connected for every nonzero ring component Re of R, then the connected components of G * ( H R ) are precisely the graphs G * ( R e ) .

On the difference of the enhanced power graph and the power graph of a finite group

The difference graph D(G) of a finite group G is the difference of the enhanced power graph of G and the power graph of G, where all isolated vertices are removed. In this paper we study the connectedness and perfectness of D(G) with respect to various properties of the underlying group G. We also find several connections between the difference graph of G and the Gruenberg-Kegel graph of G. We also examine the operation of twin reduction on graphs, a technique which produces smaller graphs which may be easier to analyze. Applying this technique to simple groups can have a number of outcomes, not fully understood, but including some graphs with large girth.

The number of spanning trees of E-extended power graphs associated with finite semigroups

We introduce a new kind of graph built from finite semigroups that we refer to as E-extended power graphs. An E-extended power graph ΓS of a finite semigroup S can be considered as the union of the power graph P(S) of S and the tree generated by E(S), where E(S) is the set of idempotents of S. Actually, ΓS is a minimal connected graph with vertex set S and containing P(S). In particular, if S is a finite semigroup with a unique idempotent we have ΓS=P(S). We give the number of spanning trees of an E-extended power graph of a finite semigroup, in particular, (Zn,⋅). Finally, as an application we discuss the number of spanning trees of an E-extended power graph of completely simple semigroups and completely regular semigroups.

Some new results concerning power graphs and enhanced power graphs of groups

The directed power graph P→(G) of a group G is the simple digraph with vertex set G such that x→y if y is a power of x. The power graph of G, denoted by P(G), is the underlying simple graph. The enhanced power graph Pe(G) of G is the simple graph with vertex set G in which two elements are adjacent if they generate a cyclic subgroup.In this paper, it is proven that, if two groups have isomorphic power graphs, then they have isomorphic enhanced power graphs, too. It is known that any finite nilpotent group of order divisible by at most two primes has perfect enhanced power graph. We investigated whether the same holds for all finite groups, and we have obtained a negative answer to that question. Further, we proved that, for any n≥0 and prime numbers p and q, every group of order pnq and p2q2 has perfect enhanced power graph. We also give a complete characterization of symmetric and alternative groups with perfect enhanced graphs. Finally, we briefly paid attention to the difference graph, which is a difference between the enhanced graph and power graph and proved that groups whose difference graphs are perfect, have perfect enhanced graphs.

Fatigue in Parkinson's Disease Is Due to Decreased Efficiency of the Frontal Network: Quantitative EEG Analysis.

Fatigue is a common, debilitating nonmotor symptom of Parkinson's disease (PD), but its mechanism is poorly understood. We aimed to determine whether electroencephalography (EEG) could objectively measure fatigue and to explore the pathophysiology of fatigue in PD. We studied 32 de novo PD patients who underwent EEG. We compared brain activity between 19 PD patients without fatigue and 13 PD patients with fatigue via EEG power spectra and graphs, including the global efficiency, characteristic path length, clustering coefficient, small-worldness, local efficiency, degree centrality, closeness centrality, and betweenness centrality. No significant differences in absolute or relative power were detected between PD patients without or with fatigue (all p > 0.02, Bonferroni-corrected). According to our network analysis, brain network efficiency differed by frequency band. Generally, the brain network in the frontal area for theta and delta bands showed greater efficiency, and in the temporal area, the alpha1 band was less efficient in PD patients without fatigue (p < 0.0001, p = 0.0011, and p = 0.0007, respectively, Bonferroni-corrected). Our study suggests that PD patients with fatigue have less efficient networks in the frontal area than PD patients without fatigue. These findings may explain why fatigue is common in PD, a frontostriatal disorder. Increased efficiency in the temporal area in PD patients with fatigue is assumed to be compensatory. Brain network analysis using graph theory is more valuable than power spectrum analysis in revealing the brain mechanism related to fatigue.

On the general position number of the k-th power graphs

A set is called a general position set of a graph G if any triple set V 0 of R is non-geodesic, that is, three elements of V 0 cannot lie on the same geodesic of G. The general position number gp(G) of a graph G is the cardinality of a largest general position set of G. The k-th power, Gk , of a graph G is obtained from G by adding a new edge between each pair of vertices with a distance of at most k in G. In this paper we establish the bounds for gp(Gk ) and and give an explicit formula of . Also we consider the relation between gp(G 2) and gp(G). We deduce a formula of and provide the upper bound on gp(G 2)−gp(G) for general block graphs G. Moreover, we determine the gp-number for the square of trees.

Forbidden subgraphs in enhanced power graphs of finite groups

Forbidden subgraphs in enhanced power graphs of finite groups

Characteristic polynomial of maximum and minimum matrix of square power graph of dihedral group of order 2n with odd natural number n

Characteristic polynomial of maximum and minimum matrix of square power graph of dihedral group of order 2n with odd natural number n

Laplacian polynomial of square power graph of dihedral group of order 2n with even natural number n

Laplacian polynomial of square power graph of dihedral group of order 2n with even natural number n

Spectral Properties of Power Graph of Dihedral Groups

This paper focuses on the power graph of dihedral groups of order $2n$, $D_{2n}$, where $n \geq 3$. We show the characteristic polynomial of the power graph corresponding to the adjacency, Laplacian, signless Laplacian, and normalized form of these matrices.

A Note on the Laplacian Energy of the Power Graph of a Finite Cyclic Group

In this study, the Laplacian matrix concept for the power graph of a finite cyclic group is redefined by considering the block matrix structure. Then, with the help of the eigenvalues of the Laplacian matrix in question, the concept of Laplacian energy for the power graphs of finite cyclic groups was defined and introduced into the literature. In addition, boundary studies were carried out for the Laplacian energy in question using the concepts the trace of a matrix, the Cauchy-Schwarz inequality, the relationship between the arithmetic mean and geometric mean, and determinant. Later, various results were obtained for the Laplacian energy in question for cases where the order of a cyclic group is the positive integer power of a prime.

On Finite Groups Whose Power Graphs are Line Graphs

On Finite Groups Whose Power Graphs are Line Graphs

Generalization of CD-Number for Power Graph of Some Special Types of Tree Graphs

Objectives: The main objective of the article is to finding the corona domination for the power graph of some special types of tree graph. Method: A dominating set of a graph is said to be a corona dominating set if every vertex in is either a pendant vertex or a support vertex. The minimum cardinality of a corona dominating set is called the corona domination number and is denoted by . Findings: In this article, we study the -number for the power of PVB-tree and where and identify their exact values. Novelty: The corona domination was one of the recently developed domination parameter, along with this parameter we found the exact value for some special types of graphs. Keywords: Domination Number, Corona Domination Number, Total Domination, Power Graphs, Olive Tree And PVB­Tree

Characteristic Polynomial of Power Graph for Dihedral Groups Using Degree-Based Matrices

A fundamental feature of spectral graph theory is the correspondence between matrix and graph. As a result of this relation, the characteristic polynomial of the graph can be formulated. This research focuses on the power graph of dihedral groups using degree-based matrices. Throughout this paper, we formulate the characteristic polynomial of the power graph of dihedral groups based on seven types of graph matrices which include the maximum degree, the minimum degree, the greatest common divisor degree, the first Zagreb, the second Zagreb, the misbalance degree, and the Nirmala matrices.