In this paper, sharp upper bounds for the domination number, total domination number and connected domination number for the Cayley graph G = Cay(D2n, Ω) constructed on the finite dihedral group D2n, and a specified generating set Ω of D2n. Further efficient dominating sets in G = Cay(D2n, Ω) are also obtained. More specifically, it is proved that some of the proper subgroups of D2n are efficie...

In the title compound, C(14)H(14)N(2)O(3), the nitro group is nearly coplanar with the benzene ring to which it is bonded [dihedral angle = 1.70 (2)°], and this ring is para-substituted by the amino-methyl-ene group. The dihedral angle between the benzene rings is 57.8 (1)°. The crystal structure is stabilized by N-H⋯O and C-H⋯O hydrogen bonds and weak C-H⋯π inter-actions are also observed.

In the title compound, C(24)H(28)N(2)O(4)S, the dihedral angle between the phenol ring and the thia-zole ring system is 10.6 (1)°, and the trimethoxy-phenyl group is approximately perpendicular to the thia-zole ring, the dihedral angle being 84.7 (2)°. There is a strong intra-molecular hydrogen-bonding inter-action between the Schiff base and the hydr-oxy group.

In this paper, we find a combinatorial identity which is related to the multi-dimensional integral γ m in study of divisor functions (see (1)). As an application, determine finite dual group algebra infinite dihedral group.