Notes on the equitable graph of type II of a finite group

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Abstract

Given a finite group G, the equitable graph of type II defined on G is an undirected graph with vertex set G, in which two distinct vertices a and b are adjacent if and only if either 0 < | o(a) − o(b)| ≤ min{o(a), o(b)} or one of a and b is equal to e, where o(a) and o(b) are the orders of a and b, respectively. In this paper, we observe that every equitable graph of type II of a finite group is a generalized lexicographic product and discuss finite groups G whose equitable graph of type II is C_n-free, where n ≥ 3. As an application, we show that every equitable graph of type II of a finite group is perfect. Finally, we characterize the metric dimension of the equitable graph of type II of a finite group.

Keywords

equitable graph of type II / finite group / metric dimension

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