Directed Power Graphs in Group Theory

• Directed power graphs are directed graphs constructed from groups, where each vertex represents an element and a directed edge from x to y signifies that y is a positive power of x.
• The structure of these graphs is analyzed through cyclic subgroup equivalence and closed-twin partitions, enabling algorithmic reconstruction with O(n²) complexity in finite groups.
• In applied settings, directed power graphs model electrical grids by utilizing weighted Laplacians to capture network connectivity and facilitate effective Kron reduction.

A directed power graph is a digraph constructed from an algebraic structure, most notably a group, by encoding the power relations among its elements as directed edges. In a directed power graph of a group $G$, each vertex represents an element of $G$, and there is an arc from $x$ to $y$ if $y$ is a positive power of $x$. This concept extends to network modeling in applied mathematics, notably in the analysis of power flow in electrical grids, where network structure and directionality are captured via appropriately defined Laplacians on directed graphs. The theory of directed power graphs bridges combinatorial, algebraic, and network-theoretic methodologies.

1. Definitions and Algebraic Foundation

Given a finite group $G$ with identity element 1, the directed power graph $P(G) = (V,A)$ is defined by:

• $V = G$
• $$A = \{(x,y) \in G^2 : x \ne y,\, y = x^m \text{ for some } m \in \mathbb{N}\}$$

An arc $x \to y$ exists if $y$ is a nontrivial positive power of $x$ (Bubboloni et al., 2022, Cameron et al., 2017, Acharyya et al., 2020). The undirected power graph has the same vertex set, with an edge $\{x, y\}$ whenever $(x, y)$ or $(y, x)$ is a directed edge.

For infinite or torsion-free groups, the directed power graph is defined analogously: $x \to y$ iff $y \in \langle x \rangle$, where $\langle x \rangle$ is the cyclic subgroup generated by $x$ (Cameron et al., 2017, Zahirović, 2020).

In applied contexts, particularly in modeling physical power networks, a directed graph $\mathcal{G}_n = (\mathcal{V}, \mathcal{E}_d)$ can represent the network, and a weighted Laplacian is used to encode the connectivity and directionality (see Section 6) (Wang et al., 2023).

2. Structure, Classes, and Reconstructions in Finite Groups

Closed Twin Equivalence and Moore Closure

To analyze the structure of power graphs, the closed neighborhood $N[v]$ of $v$ in the undirected graph is considered. The closed-twin equivalence $\sim$ identifies elements with identical closed neighborhoods, leading to $N$-classes (Bubboloni et al., 2022).

Refinement via $\langle x \rangle = \langle y \rangle$ (cyclic-subgroup equivalence, denoted $\equiv$) partitions $G$ into:

• Plain classes: a single $\equiv$-class.
• Compound classes: unions of multiple distinct $\equiv$-classes.
• Critical classes: plain or compound, subject to cardinality and closure properties, e.g., sets $C$ with $|C| = p^r$ (prime $p$, $r \ge 2$) and $C = c(C) \setminus \{1\}$, where $c$ is a Moore closure operator (Bubboloni et al., 2022).

Algorithmic Reconstruction

Bubboloni and Pinzauti provide an explicit algorithm to reconstruct the directed power graph from the undirected one, leveraging the above classifications and deterministic orientation rules:

• Edges between $\langle x \rangle$-classes are oriented from larger to smaller cardinality.
• Within $\langle x \rangle$-classes, every edge yields both arcs.
• Special handling is applied for involution classes and the unique "star" vertex (Bubboloni et al., 2022).

Correctness is formally justified and the complexity of reconstruction is $O(n^2)$ in the group order.

3. Directionality Determination and Isomorphism Classes

For finite groups, Cameron proved that the undirected power graph determines the directed power graph up to isomorphism (Cameron et al., 2017, Acharyya et al., 2020). Any isomorphism between undirected power graphs automatically preserves the orientations of every directed edge.

For infinite groups, this property is more nuanced:

• Torsion-free groups with property (*)—every nonidentity element lies in a unique maximal cyclic subgroup (e.g., free abelian groups, free groups)—also allow the directed power graph to be reconstructed from the undirected power graph.
• In such cases, each connected component corresponds canonically to a copy of the infinite cyclic group's power graph (minus identity) and orientation is determined by combinatorial tests on neighborhoods (Cameron et al., 2017, Zahirović, 2020).

The results extend to torsion-free nilpotent groups of class at most 2, subgroups of the additive group of $\mathbb{Q}$, and rational vector spaces $\mathbb{Q}^n$, where finiteness and maximality arguments force uniqueness of direction recovery from the undirected structure.

4. Metric, Path, and Cycle Properties

Fundamental combinatorial properties of directed power graphs:

• The length of a longest directed path in the directed power graph of a finite group equals the size of a largest clique in the undirected power graph (Acharyya et al., 2020).
• In cyclic groups, this length is given by $\Psi(n)$, a sum involving Euler's totient function and prime divisors.
• Directed 2-cycles occur exactly among elements generating the same cyclic subgroup, while longer chordless directed cycles are impossible due to transitivity (Acharyya et al., 2020).
• The directed distance $d_{\rm dir}(x, y)$ is determined by subgroup chain length: $d_{\rm dir}(x, y) = \ell(x) - \ell(y)$, with $\ell(x)$ the composition-series length of $\langle x \rangle$ (Acharyya et al., 2020).

These properties are reflected in quotient structures such as the cyclic-subgroup graph $C(G)$, which is a directed acyclic graph whose maximum-weight path reflects maximal directed path length in $\vec{\mathfrak{g}}(G)$.

5. Power Graphs in Network and Applied Contexts

Beyond algebraic groups, directed power graphs have application in modeling physical networks, including lossless DC power-flow in electrical grids:

• Nodes represent buses, directed edges represent oriented transmission lines with associated susceptances.
• The directed weighted Laplacian $\mathcal{L} = \mathcal{H}_0 B \mathcal{H}^T$ captures the directionality and weighting of network edges, where $\mathcal{H}$ is the incidence matrix, $\mathcal{H}_0$ is the outgoing-edge version, and $B$ is diagonal in edge weights (Wang et al., 2023).
• The power-flow equation $P_v = \mathcal{L} \theta$ maps nodal phase angles $\theta$ to net power extraction/injection.
• Structural reduction via Kron reduction uses block Schur-complements to efficiently eliminate interior nodes, maintaining the power-injection and network-theoretic properties on retained nodes. This method is justified when the retained set is reachable from the eliminated set (Wang et al., 2023).

Simulation studies confirm the effectiveness of these reductions with significant reductions in matrix size and computation time, and power-flow accuracy to machine precision (Wang et al., 2023).

6. Uniqueness, Limitations, and Open Problems

A principal uniqueness theorem asserts that for all torsion-free groups, the undirected power graph determines the directed power graph up to isomorphism, except in pathological cases involving intersection-free quasicyclic subgroups (Prüfer $p$-groups) (Zahirović, 2020). For both algebraic and applied settings, the power graph's fidelity in encoding directionality is high, provided group-theoretic obstructions are absent.

Open questions remain, notably in:

• Full characterization of those (possibly torsion) groups for which the undirected power graph determines the directed one (Zahirović, 2020, Cameron et al., 2017).
• Analysis of properties such as chordality, diameter, and acyclicity in both algebraic and network contexts (Acharyya et al., 2020).
• The interplay between the algebraic structure of $G$, closure operators, and reachability/directionality in applied directed graphs.

The directed power graph framework thus establishes a rich intersection of combinatorial, algebraic, and applied graph theory, where orientation encodes crucial algebraic or network-theoretic information, often recoverable from undirected structure but with nuanced dependence on group properties and connectivity.