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)|\le \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\ge 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.
Citation: Chunqiang Cui, Changliang Wang. Notes on the equitable graph of type II of a finite group[J]. Electronic Research Archive, 2026, 34(3): 2087-2098. doi: 10.3934/era.2026093
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)|\le \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\ge 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.
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Chunqiang Cui, Changliang Wang. Notes on the equitable graph of type II of a finite group[J]. Electronic Research Archive, 2026, 34(3): 2087-2098. doi: 10.3934/era.2026093
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