The pull-back [g.sup.*]: [N.sup.1.sub.C](Y) [equivalent] [N.sup.1.sub.C](Y) induces an
automorphism of H.
Some preliminary results about monoidal Hom-algebras with Frobenius property have been studied, such as when finite dimensional monoidal Hom-Hopf algebras are Frobenius associated with integral spaces [16], the structures of separable and Frobenius monoidal Hom-algebras related to the Hom-Frobenius-separability equation, as well as the Nakayama
automorphism of Frobenius monoidal Hom-algebras [24], and so on.
Since the Dunkl transform is an
automorphism of S([R.sup.d]), the space is a closed linear subspace of S([R.sup.d]).
This endomorphism is an
automorphism if and only if k and 5 are coprime.
Similarly we can define [mathematical expression not reproducible] is the identity
automorphism of [mathematical expression not reproducible].
Next we would determine the
automorphism group of the digraph G(H, f).
Let [GAMMA] be a graph with
automorphism group Aut([GAMMA]).
Bhutani, "On
automorphisms of fuzzy graphs," Pattern Recognition Letters, vol.
(i) If [mathematical expression not reproducible] is a finite field, then [mathematical expression not reproducible] is a cyclic group of order m and is generated by Frobenius
automorphism [[phi].sub.p].
A topological index is a numeric quantity associated with a graph which characterize the topology of graph and is invariant under graph
automorphism. There are some major classes of topological indices such as distance based topological indices, degree based topological indices and counting related polynomials and indices of graphs.