http://www.maths.qmul.ac.uk/~pjc/preprints/power5.pdf

The power graph of a finite group

This article gives a survey of all results on the power graphs of groups and semigroups obtained in the literature. Various conjectures due to other authors, questions and open problems are also included.

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Power graphs: A survey

This article surveys the recent and active area of zero-divisor graphs of commutative rings. Notable algebraic and graphical results are given, followed by a historical overview and an extensive bibliography.

Zero-divisor graphs in commutative rings

Abstract The directed power graph of a group G is the digraph with vertex set G, having an arc from y to x whenever x is a power of y; the undirected power graph has an edge joining x and y whenever one is a power of the other. We show that, for a finite group, the undirected power graph determines the directed power graph up to isomorphism. As a consequence, two finite groups which have isomorphic undirected power graphs have the same number of elements of each order.

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The power graph of a finite group, II

The structure and metric dimension of the power graph of a finite group

The undirected power graph G(S) of a semigroup S is an undirected graph whose vertex set is S and two vertices a,b∈S are adjacent if and only if a≠b and am=b or bm=a for some positive integer m. In this paper we characterize the class of semigroups S for which G(S) is connected or complete. As a consequence we prove that G(G) is connected for any finite group G and G(G) is complete if and only if G is a cyclic group of order 1 or pm. Particular attention is given to the multiplicative semigroup ℤn and its subgroup Un, where G(Un) is a major component of G(ℤn). It is proved that G(Un) is complete if and only if n=1,2,4,p or 2p, where p is a Fermat prime. In general, we compute the number of edges of G(G) for a finite group G and apply this result to determine the values of n for which G(Un) is planar. Finally we show that for any cyclic group of order greater than or equal to 3, G(G) is Hamiltonian and list some values of n for which G(Un) has no Hamiltonian cycle.

Ivy Chakrabarty

Papers

Undirected power graphs of semigroups

Intersection graphs of ideals of rings

Intersection graphs of ideals of rings