ABSTRACT

The power graph of a group Ω is a graph, whose node set is Ω and two distinct elements are adjacent if and only if one is an integral power of the other. A metric dimension of a graph Γ, denoted by $\psi \left(\mathrm{\Gamma }\right)$ is the minimum cardinality of the resolving set of Γ. In this context, we study distant properties and detour distant properties such as closure, interior, distance degree sequence and eccentric subgraphs of the power graphs of certain finite non-abelian groups. As a consequence, we figure out the metric dimension and resolving polynomial of power graphs for dihedral and generalized quaternion groups by using neighbourhood and twin sets.

中文翻译：

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在某些有限群的幂图上

摘要

组Ω的功率图是一个图，其节点集为Ω，并且当且仅当一个是另一个的整数幂时，两个不同的元素才相邻。图Γ的度量尺寸，表示为$\psi （\mathrm{\Gamma }）$是Γ的解析集合的最小基数。在这种情况下，我们研究了某些有限非阿贝尔群的远方特性和de回远方特性，例如闭合度，内部，距离度序列和偏心子图。结果，我们通过使用邻域和孪生集找出了二面角和广义四元数组的度量图的度量维数和解析多项式。