The enhanced power graph of a finite group is the simple graph with vertex set in which distinct elements and are adjacent iff they generate a cyclic group (Aalipour et al. 2017). Equivalently, they must belong to a common cyclic subgroup. In the ordinary power graph, adjacency instead requires one of and to be a power of the other.

If is a cyclic group, then is a complete graph. For a noncyclic elementary Abelian group, the maximal cliques are the subgroups of prime order. Distinct such subgroups meet only at the identity element.

Manna (2026, p. 1) classified the finite simple groups whose enhanced power graphs are chordal. They are the cyclic groups of prime order, the alternating group , the projective special linear groups for and , and the Suzuki groups for .

See also

Chordal Graph, Cyclic Group, Finite Group

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References

Aalipour, G.; Akbari, S.; Cameron, P. J.; Nikandish, R.; and Shaveisi, F. "On the Structure of the Power Graph and the Enhanced Power Graph of a Group." Elec. J. Combin. 24, P3.16, 2017. https://doi.org/10.37236/6497.Manna, P. "Chordality of Enhanced Power Graphs of Finite Simple Groups: A Resolution of a Conjecture." 3 Nov 2026. https://arxiv.org/abs/2611.05068v1.

Cite this as:

Weisstein, Eric W. "Enhanced Power Graph." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/EnhancedPowerGraph.html

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