David G. Costanzo, Mark L. Lewis, Stefano Schmidt, Eyob Tsegaye, Gabe Udell
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引用次数: 0
摘要
设 G 是一个有限群，以 G {1} 作为 Δ(G) 的顶点集，并在〈x, y〉循环时在两个顶点 x 和 y 之间画一条边，从而构造一个图 Δ(G)。设 K(G) 是由Δ(G) 沿同一元素的普遍顶点组成的集合。对于可解群 G，我们提出了 K(G) 是非微观的必要条件和充分条件。当 ∣G∣ 被两个不同的素数整除且 Δ(G) 的直径为 2 时，我们还发展了 Δ(G) 与 K(G) 之间的联系。
Characterizing finite groups whose enhanced power graphs have universal vertices
Let G be a finite group and construct a graph Δ(G) by taking G {1} as the vertex set of Δ(G) and by drawing an edge between two vertices x and y if 〈x, y〉 is cyclic. Let K(G) be the set consisting of the universal vertices of Δ(G) along the identity element. For a solvable group G, we present a necessary and sufficient condition for K(G) to be nontrivial. We also develop a connection between Δ(G) and K(G) when ∣G∣ is divisible by two distinct primes and the diameter of Δ(G) is 2.
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