Generalized Graph Compositions with Applications to Difference Graphs of Finite Groups
Abstract
The difference graph D(G) of a finite group G is obtained from the edge difference between its intersection power graph and power graph, after deleting isolated vertices. This graph has already been studied, with sufficient conditions for connectedness and a diameter bound $6$ for finite groups satisfying those conditions. We use generalized graph composition to reduce D(G) to a graph B(G) on the cyclic subgroups of G, so that connectedness and diameter are determined by the subgroup structure of G. We obtain a general criterion for the non-emptiness of B(G) in terms of branching subgroups and b-normality, and characterize its connectedness for finite p-groups, non-cyclic finite abelian groups, and non-abelian groups with both trivial and non-trivial center. Combined with the previously established cyclic-group case, this gives a complete characterization of non-emptiness and connectedness of difference graphs for all finite groups. The successive structural cases lead naturally to the sharp diameter bounds $2,3,4,$ and $5$. For centerless non-abelian groups, connectedness is governed either by a unique branching subgroup or by an auxiliary graph 𝒜(G); in the latter case diam𝒜(G)−1 ≤ diamB(G) ≤ max{4,diam𝒜(G)+1}, and both bounds are sharp.
Related
- A Generalized Finite Difference-Based Fragile Points Method for Heat Conduction Problems in Non-Homogeneous Media
- An Equivariant Main Conjecture for Branched ℤp-Towers of Finite Graphs and Applications
- Initial Runs in Integer Compositions: Lambert Series and Generalized Divisor Sums
- Monogenic Fields from Polynomial Compositions with Applications
- Generalized linear cellular automata in groups and difference Galois theory II
BibTeX
@misc{ghosh2026generalized,
title = {{Generalized Graph Compositions with Applications to Difference Graphs of Finite Groups}},
author = {Shamik Ghosh and Sanchita Paul and M. K. Sen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.00080},
eprint = {2609.00080},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00080},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Generalized Graph Compositions with Applications to Difference Graphs of Finite Groups
AU - Shamik Ghosh
AU - Sanchita Paul
AU - M. K. Sen
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.00080
UR - https://arxiv.org/abs/2609.00080
AB - The difference graph D(G) of a finite group G is obtained from the edge difference between its intersection power graph and power graph, after deleting isolated vertices. This graph has already been studied, with sufficient conditions for connectedness and a diameter bound $6$ for finite groups satisfying those conditions. We use generalized graph composition to reduce D(G) to a graph B(G) on the cyclic subgroups of G, so that connectedness and diameter are determined by the subgroup structure of G. We obtain a general criterion for the non-emptiness of B(G) in terms of branching subgroups and b-normality, and characterize its connectedness for finite p-groups, non-cyclic finite abelian groups, and non-abelian groups with both trivial and non-trivial center. Combined with the previously established cyclic-group case, this gives a complete characterization of non-emptiness and connectedness of difference graphs for all finite groups. The successive structural cases lead naturally to the sharp diameter bounds $2,3,4,$ and $5$. For centerless non-abelian groups, connectedness is governed either by a unique branching subgroup or by an auxiliary graph 𝒜(G); in the latter case diam𝒜(G)−1 ≤ diamB(G) ≤ max{4,diam𝒜(G)+1}, and both bounds are sharp.
ER -