The power graph P(G) of a group G is the graph whose vertex set is the group elements and two elements are adjacent if one is a power of the other. In this paper, we consider some graph theoretical properties of a power graph P(G) that can be related to its group theoretical properties. As consequences of our results, simple proofs for some earlier results are presented.

Typically, a large-scale natural graph follows a skewed power law. In distributed graphstructured computations, the skewness usually makes a bad partitioning, which leads to high communication cost and workload imbalance. Therefore, graph partitioning (GP) is a challenging issue. To tackle this challenge, we introduce degree-based techniques into GP via vertex-cut. Accordingly, we develop a nov...

We deal with the graph operator Pow2 defined to be the complement of the square of a graph: Pow2(G) = Pow2(G). Motivated by one of many open problems formulated in [6] we look for graphs that are 2-periodic with respect to this operator. We describe a class G of bipartite graphs possessing the above mentioned property and prove that for any m,n ≥ 6, the complete bipartite graph Km,n can be deco...

The power graph of a group is the graph whose vertex set is the group, two elements being adjacent if one is a power of the other. We observe that nonisomorphic finite groups may have isomorphic power graphs, but that finite abelian groups with isomorphic power graphs must be isomorphic. We conjecture that two finite groups with isomorphic power graphs have the same number of elements of each o...

Context-free hypergraph grammars and boundary graph grammars of bounded nonterminal degree have the same power, both for generating sets of graphs and for generating sets of hypergraphs. Arbitrary boundary graph grammars have more graph generating power than context-free hypergraph grammars, but they have the same hypergraph generating power. To obtain these results, several normal forms for bo...

The NP-complete Closest 4-Leaf Power problem asks, given an undirected graph, whether it can be modified by at most r edge insertions or deletions such that it becomes a 4-leaf power. Herein, a 4-leaf power is a graph that can be constructed by considering an unrooted tree—the 4-leaf root—with leaves one-to-one labeled by the graph vertices, where we connect two graph vertices by an edge iff th...

We consider cooperative transferable utility games, or simply TU-games, with limited communication structure in which players can cooperate if and only if they are connected in the communication graph. Solutions for such graph games can be obtained by applying standard solutions to a modified or restricted game that takes account of the cooperation restrictions. We discuss Harsanyi solutions wh...

Due to the limited resources available in the wireless ad hoc networking nodes, the scalability is crucial for network operations. One effective approach is to maintain only a sparse spanner of a linear number of links while still preseving the power-efficient route for any pair of nodes. For any spanner G, its power stretch factor is defined as the maximum ratio of the minimum power needed to ...

The underlying theme of this paper is to explore the various facets power systems data through lens graph signal processing (GSP), laying down foundations Grid-GSP framework. provides an interpretation for spatio-temporal properties voltage phasor measurements, by showing how well-known modeling supports a generative low-pass filter model state variables, namely phasors. Using we formalize empi...