On embeddings of the difference graph of the intersection power graph and the power graph
Abstract
The power graph of a finite group G is a simple undirected graph with vertex set G and two vertices are adjacent if one is a power of the other. The intersection power graph of a finite group G is a simple undirected graph with vertex set G and two vertices x, y are adjacent if ⟨ x⟩ ∩ ⟨ y ⟩ ≠ {e}. The difference graph 𝒟(G) of a finite group G is the difference of the intersection power graph and power graph with all isolated vertices removed. We characterized all the finite nilpotent groups G except $2$-group such that the difference graph is planar. Further, we determine all the finite nilpotent groups whose difference graph has genus at most $2$. Moreover, we prove that there does not exist any finite group whose difference graph is projective planar.
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BibTeX
@misc{manisha2026embeddings,
title = {{On embeddings of the difference graph of the intersection power graph and the power graph}},
author = {Manisha and Ekta and Jitender Kumar},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.02945},
eprint = {2609.02945},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02945},
note = {CC0},
}
RIS
TY - UNPB
TI - On embeddings of the difference graph of the intersection power graph and the power graph
AU - Manisha
AU - Ekta
AU - Jitender Kumar
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2609.02945
UR - https://arxiv.org/abs/2609.02945
AB - The power graph of a finite group G is a simple undirected graph with vertex set G and two vertices are adjacent if one is a power of the other. The intersection power graph of a finite group G is a simple undirected graph with vertex set G and two vertices x, y are adjacent if ⟨ x⟩ ∩ ⟨ y ⟩ ≠ {e}. The difference graph 𝒟(G) of a finite group G is the difference of the intersection power graph and power graph with all isolated vertices removed. We characterized all the finite nilpotent groups G except $2$-group such that the difference graph is planar. Further, we determine all the finite nilpotent groups whose difference graph has genus at most $2$. Moreover, we prove that there does not exist any finite group whose difference graph is projective planar.
ER -