Power graph of a group G, represented by \Gamma_G, is a graph where the vertex set consists of the elements of G. Two distinct vertices a, b \in G are connected by an edge if and only if there exists a positive integer m such that a^m = b or b^m = a. This study explores the utilization of a new approach to compute the topological indices of power graph associated with dihedral group with n=p^k, p is primes and k \in \mathbb{Z}. Results obtained indicate that the topological indices calculated using new approach yield the same values as those obtained with the conventional approach.

The power graph [Formula: see text] of a group [Formula: see text] is the simple and undirected graph with vertex set [Formula: see text] and with two distinct vertices being adjacent, whenever one of them is a positive power of the other in [Formula: see text]. The independence complex of a graph [Formula: see text], is the simplicial complex [Formula: see text] with the vertex set being that of [Formula: see text] and the simplices being the independent sets of [Formula: see text]. In this paper, we study the homotopy type of the independence complex of power graphs of cyclic groups of order [Formula: see text], where [Formula: see text] and [Formula: see text] are distinct primes and [Formula: see text].

The perfect state transfer and the success probability of finding the marked vertex on the N-fold star power graph G⋆N

In this paper, we investigate the structural and combinatorial properties of the kth power graph Γk(G) associated with a finite group G, where k ≥ 2. The graph Γk(G) is defined by taking the elements of G as vertices and connecting two distinct vertices x and y by an edge if either x = yk or y = xk. This construction generalizes the well-studied power graph of a group and provides new insight into the influence of exponentiation on group elements when viewed through graph-theoretical properties. We show that Γk(G) is a subgraph of the power graph P(G) and analyze conditions under which Γk(G) is connected, disconnected, or empty. Depending on the algebraic structure of G and the arithmetic properties of k, we show that Γk(G) can exhibit a variety of structural forms, including being a tree, a union of disjoint stars, or a complete multipartite graph. For instance, when G = Zn and gcd(k, n) = 1, Γk(G) decomposes into disjoint stars, while for certain non-cyclic groups, the graph becomes multipartite. Additionally, we provide formulas for computing the number of edges in Γk(G) and discuss how subgroup structure and group automorphisms impact the topology of the graph.

Let Aex(G) be the extended adjacency matrix of G. The eigenvalues of Aex(G) are called extended adjacency eigenvalues of G. The sum of the absolute values of eigenvalues of the Aex-matrix is called the extended adjacency energy Eex(G) of G. In this paper, we obtain the Aex-spectrum of the joined union of regular graphs in terms of their adjacency spectrum and the eigenvalues of an auxiliary matrix. Consequently, we derive the Aex-spectrum of the join of two regular graphs, the lexicographic product of regular graphs, and the Aex-spectrum of various families of graphs. Further, as applications of our results, we construct infinite classes of infinite families of extended adjacency equienergetic graphs. We show that the Aex-energy of the join of two regular graphs is greater than or equal to their energy. We also determine the Aex-eigenvalues of the power graph of finite abelian groups.

On the Power Graphs of Homogeneous Completely Simple Semigroups

The dihedral group is a mathematical structure generated by rotational and reflection symmetries. In this study, the representation of the group is described using a power graph, where all elements of the group are treated as vertices, and two distinct elements are considered adjacent when one is a power of the other. By analyzing the structural patterns of the resulting power graphs, various connectivity indices can be determined, particularly for dihedral groups whose orders are powers of a prime number. This research focuses on six specific connectivity indices: the first Zagreb index, the second Zagreb index, the Wiener index, the hyper-Wiener index, the Harary index, and the Szeged index.

Grothendieck group of the Leavitt path algebra over power graphs of prime-power cyclic groups

On the domination number of proper power graphs of finite groups

In this paper, we investigate the directed power graph of the direct product of cyclic groups, Zm and Zn where m and n are not relatively prime. We conduct a detailed structural analysis of the graphs G(Zp×p), G(Zp×2p)and G(Zp×p2). Here we are utilizing the algebraic properties of Zn and Zm, where m and n are coprime. The study focuses on how the lack of coprimality influences the connectivity and hierarchical structure of these directed power graphs.

Power graphs of Brandt semigroups

Abstract Some years ago, Shitov proved that the chromatic number of the power graph of semi-groups is at most countable, thus answering a question raised by Aalipour et al. about whether this statement holds for groups. Later, Dalal and Kumar proved that the chromatic number of the enhanced power graph of a group is also countable. In the line of these recent works, we first generalize the concept of power graphs to coprime power graph, and then we prove the same type of result for this generalization. Furthermore, we state a conjecture about their common relationship as graphs.

ABSTRACT Graph energy has been the main concern of spectral graph theory in the last five decades. The classical graph energy is the sum of the absolute values of the eigenvalues of the adjacency matrix. In many research papers, different versions of graph energy by utilizing different graph matrices are introduced. For many graph types corresponding to molecular structures, the energy is determined. The theory is complete for complete bipartite graphs. For derived graphs, the problem was settled partially for line, total, double and subdivision graphs. In this paper, the more complex cases of power graphs, shadow, image and core graphs are discussed, and the adjacency matrices of these derived graph classes are formed in terms of very simple submatrices.

Abstract We generalize the enhanced power graph by replacing elements with conjugacy classes. The main result of this paper is to determine when this graph is triangle-free.

A power graph is a simple, connected graph defined on a finite group, which accepts the elements of the group as vertices and defines the adjacency relationship between two vertices as “if one vertex is a power of the other”. The concept of power graph was first introduced to the mathematical literature by Kelarev and Quinn in 2000 [1]. Later, with the article of Chakrabarty et al. in 2009, studies in the field of power graphs gained momentum [2]. Another important study that shapes the power graph field is the one conducted by Chattopadhyay et al. in 2018 [3]. In their study, the adjacency matrix of the power graph was redefined by considering the block matrix structure. On the other hand, the concept of the Randić index was introduced by Milan Randić in 1975 [4]. In his study, the Randić index concept was used to determine the degree of branching of the carbon atomic skeleton. Later, with the studies carried out 1990’s the concept of Randić index entered the mathematical literature and began to be used frequently in this field. In this study, bounds were obtained for the Randić index of power graphs by considering the concepts of maximum degree, minimum degree, and vertex energy.

The utilization of wind energy as one of the renewable energy sources in Indonesia is still limited. This is due to Indonesia's large wind potential, but it is at low speeds. One modification of wind turbines to overcome this problem is by using inverse taper and taper blades. Inverse taper blades have a chord distribution that increases from root to tip while taper blades have a chord distribution that decreases from root to tip. This research conducts the design and construction of inverse taper and taper blades on small-scale wind turbines using NACA 6412 airfoil type, then tested in the field and obtained the performance of each blade. The design was carried out using blade element momentum theory to obtain the blade geometry shape and perform performance analysis of each type of blade. Based on the simulation results obtained from QBlade, it shows that the inverse taper blade performance has a maximum Cp of 0.52 while the taper blade has a maximum Cp of 0.41. The power graph from field testing results shows that inverse taper blades provide good power generation at low wind speeds compared to taper blades. The inverse taper blade has a cut-in speed of 1.2 m/s and the best power production occurs at speeds of 1-4 m/s.

Power graphs have been extensively studied for their ability to represent algebraic structures through graph-theoretic concepts. This paper investigates the structural properties of power graphs associated with B-algebras, a class of algebras that exhibit certain group-like characteristics. Several graph-theoretic properties, including graph distance measures, are examined. In addition, conditions under which the power graph is complete, Eulerian, or Hamiltonian, as well as the behavior of power graphs under B-homomorphisms, are explored. Finally, the relationship between the center of a B-algebra and the center of its corresponding power graph is established.

Sombor index is a new geometric background of graph invariants and is also called a valencybased topological descriptor. It is computed by taking the radical of the sum of the squared degrees of two adjacent vertices within a graph. The Sombor polynomial also involves the degrees of two adjacent vertices where its first order derivative at x is one, is equal to the Sombor index. Meanwhile, in a power graph, two different vertices are connected by an edge if and only if one is the power of the other. The graph’s vertex set consists of all the elements in a group. In this study, the Sombor index and Sombor polynomial of the power graph for some finite non-abelian groups are determined by using their definitions. The dihedral, generalized quaternion, and quasi-dihedral groups are considered. The generalization of the power graph for the quasi-dihedral groups are also found.

Enhanced Power Graph of Non-Abelian Group of Order p3 of Exponent p

Purpose. Study of thermal processes of an inverter based on an IGBT module for used in a frequency converter to control the operation of an asynchronous motor. Methodology. Analytical and computational methods to analyse thermal processes of an inverter based on an IGBT module. Findings. The study of thermal processes of the SKM200GB12T4 inverter based on the IGBT module was performed using the SemiSel program. A mathematical model of the cooling process of the SKM200GB12T4 inverter was developed. The dependence of the dynamic thermal impedance Zth(s-a) on time, which is described by an exponential function, was obtained. The value of the time constant for this dependence, which characterizes the rate of change in the cooler temperature, i.e. the quality of its operation, has been calculated. The thermal time constant τ = 1.44 s indicates the time required to reach a temperature difference of approximately 63% of its stationary value. This low value reflects the effective cooling due to the high air flow velocity (7 m/s) and air flow rate (426.43 m³/h), which is critically important for maintaining the IGBT junction temperature below 175 °C during overload. The values of the inverter temperature maxima during overload were obtained. For an overload of 10.94 seconds, the maximum temperature for IGBT transistors is 120.85 °C, and for diodes – 123.4 °C. The case temperature Tc = 71.21 °C and the radiator temperature Ts = 63.56 °C remain the same for transistors and diodes and do not exceed the maximum operating temperature of the module due to the stability of the cooling system. However, overheating can increase with prolonged loading, resulting in the degradation of semiconductor devices. The temperature and power variation processes at nominal load and in overload mode for one period have been studied using the SemiSel program. The temperature change graphs reflect the stability of the temperature at various points, such as the transitions of IGBT transistors and reverse diodes, due to effective thermal control. The power graph indicates cyclical changes in losses, with peaks in the phases where current and voltage are maximum. These data confirm the suitability of the module for use in control circuits. Originality. Based on the graphical analysis of the kinetic dependencies of temperature and inverter power, a mathematical model of the cooling process of the SKM200GB12T4 inverter was developed, that describes the dependence of the dynamic thermal impedance Zth(s-a) on time. The thermal time constant for this dependence, which characterises the rate of change of the cooler temperature, was calculated. Practical value. The results of the study of the thermal characteristics of the SKM200GB12T4 inverter can be used to optimize the operating modes of the frequency converter for controlling the operation of an asynchronous motor.