On the tree structure of the power digraphs modulo n

For any two positive integers n and k ? 2, let G(n, k) be a digraph whose set of vertices is {0, 1, …, n ? 1} and such that there is a directed edge from a vertex a to a vertex b if a ^k ≡ b (mod n). Let \(n = \prod\nolimits_{i = 1}^r {p_i^{{e_i}}} \) be the prime factorization of n. Let P be the set of all primes dividing n and let P ₁, P ₂ ? P be such that P ₁ ∪ P ₂ = P and P ₁ ∩ P ₂ = ?. A fundamental constituent of G(n, k), denoted by \(G_{{P_2}}^*(n,k)\), is a subdigraph of G(n, k) induced on the set of vertices which are multiples of \(\prod\nolimits_{{p_i} \in {P_2}} {{p_i}} \) and are relatively prime to all primes q ∈ P ₁. L. Somer and M. K?i?ek proved that the trees attached to all cycle vertices in the same fundamental constituent of G(n, k) are isomorphic. In this paper, we characterize all digraphs G(n, k) such that the trees attached to all cycle vertices in different fundamental constituents of G(n, k) are isomorphic. We also provide a necessary and sufficient condition on G(n, k) such that the trees attached to all cycle vertices in G(n, k) are isomorphic.     

Synthesis and characterization of active cuprous oxide particles and their catalytic application in 1,2,3‐triazole synthesis via alkyne‐azide cycloaddition reaction in water

Active cuprous oxide materials are synthesized from CuSO₄.5H₂O using sodium stannite as reducing agent in the presence of various stabilizers, viz., cetyl trimethyl ammonium bromide, sodium dodecyl sulphate, and polyvinyl pyrrolidone. The synthesized cuprous oxide materials are well characterized by powder X‐ray diffraction and Fourier transform infrared spectroscopy to ascertain their identity, while field emission scanning electron microscopy and energy‐dispersive spectroscopy analysis were used to study their morphology and composition, respectively. We have compared the catalytic prowess of the various cuprous oxide materials in the cycloaddition reaction of alkynes and azides to synthesize 1,4‐disubstituted‐1,2,3‐triazoles. A wide variety of substitutions can nicely be tolerated in our optimized reaction conditions to produce very good to excellent yields of the corresponding triazoles in water at 55 °C. The reactions are carried out in water without any assistance of organic cosolvent or other additives, which renders the catalytic method as economical and environment friendly.