Power Graph Research Articles

Distance Laplacian spectra of various graph operations and its application to graphs on algebraic structures

In this paper, we determine the distance Laplacian spectra of graphs obtained by various graph operations. We obtain the distance Laplacian spectrum of the join of two graphs [Formula: see text] and [Formula: see text] in terms of adjacency spectra of [Formula: see text] and [Formula: see text]. Then we obtain the distance Laplacian spectrum of the join of two graphs in which one of the graphs is the union of two regular graphs. Finally, we obtain the distance Laplacian spectrum of the generalized join of graphs [Formula: see text], where [Formula: see text], in terms of their adjacency spectra. As applications of the results obtained, we have determined the distance Laplacian spectra of some well-known classes of graphs, namely the zero divisor graph of [Formula: see text], the commuting and the non-commuting graph of certain finite groups like [Formula: see text] and [Formula: see text], and the power graph of various finite groups like [Formula: see text], [Formula: see text] and [Formula: see text]. We show that the zero divisor graph and the power graph of [Formula: see text] are distance Laplacian integral for some specific [Formula: see text]. Moreover, we show that the commuting and the non-commuting graph of [Formula: see text] and [Formula: see text] are distance Laplacian integral for all [Formula: see text].

The power graph of a torsion-free group of nilpotency class 2

The directed power graph $$\vec {\mathcal {G}}( G)$$ of a group G is the simple digraph with vertex set G in which $$x\rightarrow y$$ if y is a power of x, and the power graph is the underlying simple graph $$\mathcal {G}( G)$$ . In this paper, three versions of the definition of the power graph are discussed, and it is proved that the power graph by any of the three versions of the definition determines the other two up to isomorphism. It is also proved that if G is a torsion-free group of nilpotency class 2 and if H is a group such that $$\mathcal {G}( H)\cong \mathcal {G}( G)$$ , then G and H have isomorphic directed power graphs, which was an open problem proposed by Cameron, Guerra and Jurina [9].

Characterizing finite nilpotent groups associated with a graph theoretic equality

The power graph of a group is the simple graph whose vertices are the group elements and two vertices are adjacent whenever one of them is a positive power of the other. In this paper, we characterize the finite nilpotent groups whose power graphs have equal vertex connectivity and minimum degree.

Finite groups with the same power graph

The power graph P(G) of a group G is a graph with vertex set G, where two vertices u and v are adjacent if and only if and or for some positive integer m. In this paper, we raise and study the following question: For which natural numbers n every two groups of order n with isomorphic power graphs are isomorphic? In particular, it is proved that all such odd number n are cube-free and also they are not multiples of 16 in general. Moreover, we show that if two finite groups have isomorphic power graphs and one of them is nilpotent, the same is true for the other one.

The Least Eigenvalue of the Complement of the Square Power Graph of G

Let G n , m represent the family of square power graphs of order n and size m , obtained from the family of graphs F n , k of order n and size k , with m ≥ k . In this paper, we discussed the least eigenvalue of graph G in the family G n , m c . All graphs considered here are undirected, simple, connected, and not a complete K n for positive integer n .

The power graph of a torsion-free group determines the directed power graph

The directed power graph G→(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 G(G), is the underlying simple graph.In this paper, for groups G and H, the following is proved. If G has no quasicyclic subgroup Cp∞ which has trivial intersection with every cyclic subgroup K of G such that K⁄≤Cp∞, then G(G)≅G(H) implies G→(G)≅G→(H). In particular, any two torsion-free groups having isomorphic power graphs have isomorphic directed power graphs.

Graceful labeling of power graph of group Z2k−1 × Z4

The power graph of a finite group G is a special type of undirected simple graph whose vertex set is set of elements of G, in which two distinct vertices of G are adjacent if one is the power of other. Let [Formula: see text] be a finite abelian 2-group of order [Formula: see text] where [Formula: see text]. In this paper, we establish that the power graph of finite abelian group G always has graceful labeling without any condition on [Formula: see text].

Parameters of the coprime graph of a group

‎There are many different graphs one can associate to a group‎. ‎Some examples are the well-known Cayley graph‎, ‎the zero divisor graph (of a ring)‎, ‎the power graph‎, ‎and the recently introduced coprime graph of a group‎. ‎The coprime graph of a group $G$‎, ‎denoted $Gamma_G$‎, ‎is the graph whose vertices are the group elements with $g$ adjacent to $h$ if and only if $(o(g),o(h))=1$‎. ‎In this paper we calculate the independence number of the coprime graph of the dihedral groups‎. ‎Additionally‎, ‎we characterize the groups whose coprime graph is perfect‎.

B-Chromatic Sum and b-Continuity Property of Some Graphs

A b-coloring of a graph [Formula: see text] is a proper coloring of the vertices of [Formula: see text] such that there exist a vertex in each color class joined to at least one vertex in each other color classes. The b-chromatic number of a graph [Formula: see text], denoted by [Formula: see text], is the largest integer [Formula: see text] such that [Formula: see text] has a b-coloring with [Formula: see text] colors. The b-chromatic sum of a graph [Formula: see text], denoted by [Formula: see text], is introduced and it is defined as the minimum of sum of colors [Formula: see text] of [Formula: see text] for any [Formula: see text] in a b-coloring of [Formula: see text] using [Formula: see text] colors. A graph [Formula: see text] is b-continuous, if it admits a b-coloring with [Formula: see text] colors, for every [Formula: see text]. In this paper, the [Formula: see text]-continuity property of corona of two cycles, corona of two star graphs and corona of two wheel graphs with unequal number of vertices is discussed. The b-continuity property of corona of any two graphs with same number of vertices is also discussed. Also, the b-continuity property of Mycielskian of complete graph, complete bipartite graph and paths are discussed. The b-chromatic sum of power graph of a path is also obtained.

Finite groups whose intersection power graphs are toroidal and projective-planar

Abstract The intersection power graph of a finite group G G is the graph whose vertex set is G G , and two distinct vertices x x and y y are adjacent if either one of x x and y y is the identity element of G G , or ⟨ x ⟩ ∩ ⟨ y ⟩ \langle x\rangle \cap \langle y\rangle is non-trivial. In this paper, we completely classify all finite groups whose intersection power graphs are toroidal and projective-planar.

Bi-fluid cooling effect on electrical characteristics of flexible photovoltaic panel

A photovoltaic (PV) system integrated with a bi-fluid cooling mechanism, which is known as photovoltaic thermal (PVT) system, was investigated. The electrical characteristics of flexible solar panel were evaluated for PV and PV with bi-fluid (air and water) cooling system. The integration of monocrystalline flexible solar panel into both systems was tested under a fixed solar radiation of 800 W/m2. A total of 0.04–0.10 kg/s of air flow was utilised in PV with cooling system with a fixed water mass flow rate of 0.025 kg/s. The efficiencies of flexible panel for PV and PV with cooling system were explored. For PV with bi-fluid flow, the highest obtained efficiency of module was 15.95% when 0.08 kg/s of air and 0.025 kg/s of water were allowed to flow through the cooling system. Compared with PV without cooling mechanism, the highest efficiency of module was 13.35% under same solar radiation. Current–voltage and power graphs were also plotted to present the electrical characteristics (current, voltage and power) generated by both systems.

Leavitt path algebras for power graphs of finite groups

The aim of this paper is the characterization of algebraic properties of Leavitt path algebra of the directed power graph [Formula: see text] and also of the directed punctured power graph [Formula: see text] of a finite group [Formula: see text]. We show that Leavitt path algebra of the power graph [Formula: see text] of finite group [Formula: see text] over a field [Formula: see text] is simple if and only if [Formula: see text] is a direct sum of finitely many cyclic groups of order 2. Finally, we prove that the Leavitt path algebra [Formula: see text] is a prime ring if and only if [Formula: see text] is either cyclic [Formula: see text]-group or generalized quaternion [Formula: see text]-group.

Forbidden Subgraphs of Power Graphs

The undirected power graph (or simply power graph) of a group $G$, denoted by $P(G)$, is a graph whose vertices are the elements of the group $G$, in which two vertices $u$ and $v$ are connected by an edge between if and only if either $u=v^i$ or $v=u^j$ for some $i$, $j$.
 A number of important graph classes, including perfect graphs, cographs, chordal graphs, split graphs, and threshold graphs, can be defined either structurally or in terms of forbidden induced subgraphs. We examine each of these five classes and attempt to determine for which groups $G$ the power graph $P(G)$ lies in the class under consideration. We give complete results in the case of nilpotent groups, and partial results in greater generality. In particular, the power graph is always perfect; and we determine completely the groups whose power graph is a threshold or split graph (the answer is the same for both classes). We give a number of open problems.

Directed Power Graphs

In this article, first we introduce six types of power graphs related to a graph (or directed graph), with the help of set theory.Then we show that these newly defined power graphs are pairwise distinct by a few examples. Finally, we discuss the relation between Eulerian being the base graph and these six power graph types. Moreover, we express the relation between pairwise Eulerian of these power graphs.

Acceptance Ratio Analysis: An Early Fault Indicator for Grid-Connected Photovoltaic System

This paper presents the Acceptance Ratio (AR) analysis for three different grid-connected photovoltaic (GCPV) systems working under the Malaysia tropical climate. AR is a ratio between actual AC power, PAC_actual, and predicted AC power, PAC_predicted. According to Malaysian Standard MS2692:2020,the AR value must ≥ 0.9 to classify as accepted in testing and commissioning test. In contrast, a rate < 0.9 indicates a non-accepted GCPV system. Historical data of the AC power output, solar irradiance, and module temperature from January 1 to December 31, 2019, were used for the analysis. The analysis procedure was carried out using Matlab and Microsoft Excel software. The analysis covers the AC power analysis and the AR analysis based on the threshold of 0.9. The plotted monthly AC power graph shows that all systems have lower than 15 % differences between actual and predicted AC power. On the AR analysis, System 1 was found to show early fault indicator with a monthly cumulative percentage of AR < 0.9 ranges from 34 % to 71 %, meanwhile System 2 and System 3 have a lower cumulative percentage of AR < 0.9 ranges from 5 % to 19 %. This result suggested that only System 2 and 3 are fault-free GCPV systems and working in good condition. The outcome of this study has succeeded in providing preliminary AR analysis results for three GCPV systems located in Malaysia. This study would help to evaluate AR threshold reliability to indicate an early fault of a GCPV system.

On the structure of finite groups with dominatable enhanced power graph

Let [Formula: see text] be a group. The enhanced power graph of [Formula: see text] is a graph with vertex set [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if there exists [Formula: see text] such that [Formula: see text] and [Formula: see text] for some [Formula: see text]. Also, a vertex of a graph is called dominating vertex if it is adjacent to every other vertex of the vertex set. Moreover, an enhanced power graph is said to be a dominatable graph if it has a dominating vertex other than the identity element. In an article of 2018, Bera and his coauthor characterized all abelian finite groups and nonabelian finite [Formula: see text]-groups such that their enhanced power graphs are dominatable (see [2]). In addition as an open problem, they suggested characterizing all finite nonabelian groups such that their enhanced power graphs are dominatable. In this paper, we try to answer their question. We prove that the enhanced power graph of finite group [Formula: see text] is dominatable if and only if there is a prime number [Formula: see text] such that [Formula: see text] and the Sylow [Formula: see text]-subgroups of [Formula: see text] are isomorphic to either a cyclic group or a generalized quaternion group.

ON THE AUTOMORPHISMS GROUP OF FINITE POWER GRAPHS

The power graph of a group $G$ is the graph with vertex set $G$,having an edge joining $x$ and $y$ whenever one is a power of theother. The purpose of this paper is to study the automorphismgroups of the power graphs of infinite groups.

Strong metric dimensions for power graphs of finite groups

Let G be a finite group. The order supergraph of G is the graph with vertex set G, and two distinct vertices x, y are adjacent if or The enhanced power graph of G is the graph whose vertex set is G, and two distinct vertices are adjacent if they generate a cyclic subgroup. The reduced power graph of G is the graph with vertex set G, and two distinct vertices x, y are adjacent if or In this article, we characterize the strong metric dimension of the order supergraph, the enhanced power graph and the reduced power graph of a finite group.

Spectrum of proper power graphs of the direct product of certain finite groups

The proper power graph of a group G is the simple graph with non-trivial elements of G as vertices and two distinct elements are adjacent if and only if one is a power of the other. The aim of this paper is to find the structure and to compute the spectrum of the proper power graph of direct product of two finite groups and where at least one of and is an EPO group. Also we provide the full spectrum and hence the energy of whenever both and are EPO groups.

Laplacian spectra of power graphs of certain prime-power Abelian groups

The power graph [Formula: see text] of a semigroup [Formula: see text] is a simple undirected graph whose vertex set is [Formula: see text] itself, and any two distinct vertices are adjacent if one of them is a power of the other. In this paper, we describe the power graph [Formula: see text] in terms of joins and disjoint unions of complete graphs, and use this to calculate the Laplacian polynomial of [Formula: see text]. We finally calculate the Laplacian polynomial of the power graph [Formula: see text].