On the uniqueness of the factorization of power digraphs modulo n
Rendiconti del Seminario Matematico della Università di Padova, Tome 140 (2018), pp. 185-219.

For each pair of integers n= i=1 r p i e i and k2, a digraph G(n,k) is one with vertex set {0,1,...,n-1} and for which there exists a directed edge from x to y if x k y(modn). Using the Chinese Remainder Theorem, the digraph G(n,k) can be written as a direct product of digraphs G(p i e i ,k) for all i such that 1ir. A fundamental constituent G P * (n,k), where PQ={p 1 ,p 2 ,...,p r }, is a subdigraph of G(n,k) induced on the set of vertices which are multiples of p i P p i and are relatively prime to all primes p j QP. In this paper, we investigate the uniqueness of the factorization of trees attached to cycle vertices of the type 0, 1, and (1,0), and in general, the uniqueness of G(n,k). Moreover, we provide a necessary and sufficient condition for the isomorphism of the fundamental constituents G P * (n,k 1 ) and G P * (n,k 2 ) of G(n,k 1 ) and G(n,k 2 ) respectively for k 1 k 2 .

Publié le :
DOI : 10.4171/RSMUP/8
Classification : 05, 11
Mots-clés : Power digraph, direct product, uniqueness of factorization, Chinese remainder theorem
Sawkmie, Amplify 1 ; Singh, Madan Mohan 1

1 North-Eastern Hill University, Shillong, Meghalaya, India
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     title = {On the uniqueness of the factorization of power digraphs modulo $n$},
     journal = {Rendiconti del Seminario Matematico della Universit\`a di Padova},
     pages = {185--219},
     publisher = {European Mathematical Society Publishing House},
     address = {Zuerich, Switzerland},
     volume = {140},
     year = {2018},
     doi = {10.4171/RSMUP/8},
     mrnumber = {3880217},
     zbl = {1410.05082},
     url = {http://www.numdam.org/articles/10.4171/RSMUP/8/}
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Sawkmie, Amplify; Singh, Madan Mohan. On the uniqueness of the factorization of power digraphs modulo $n$. Rendiconti del Seminario Matematico della Università di Padova, Tome 140 (2018), pp. 185-219. doi : 10.4171/RSMUP/8. http://www.numdam.org/articles/10.4171/RSMUP/8/

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