# On the structure of isoclinism classes of the non-commuting graphs

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## Abstract

In this paper, we introduce the new concept isoclinism of two non-commuting graphs. We describe it with this hope to determine the properties of the graph with large number of vertices and edges more easier by use of its smaller correspondence graph in its isoclinic class. In 1939, Hekster classified the groups by *n*-isoclinism which was weaker than isomorphism, where *n* is a positive integer. The abelian groups are in the same class by group-isoclinism, although their intrinsic properties are not the same. The notion of isoclinic groups is the inspiration to define the isoclinism of two graphs. The isoclinism of two graphs is a pair of significant special isomorphism between the quotient graphs of the given graphs. We observe that all complete 3-partite non-commuting graphs are in the same isoclinic class and the non-commuting graph associated to the dihedral group of order 8, \(D_{8}\) is the representative of this class.

## Keywords

Isoclinism Non-commuting graph Quotient graph## Mathematics Subject Classification

Primary 05C25 Secondary 20B05## References

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