### Number of irreducible factors and degree in divisor graph of Zp[x,n]

#### Abstract

The divisor graph, denoted by D(Z

_{p}[x,n]), is the graph whose vertex set is the set of all polynomials of degree at most n whose coefficients are from field Z_{p}and its any two distinct vertices are adjacent if one is a divisor of the other. In this paper, (i) we determine the degree of each vertex of D(Z_{p}[x,3]) and also discuss its girth, size, degree sequence, irregularity index etc. (ii) We also establish that two polynomials of same degree k in Z_{p}[x,n] having different number of irreducible factors, the one with fewer number of irreducible factors has smaller degree. (iii) Further, if two polynomials of same degree k in Z_{p}[x,n] having same number of irreducible factors but different number of distinct irreducible factors, the one with fewer number of distinct irreducible factors has smaller degree.**Published:**2021-02-05

**How to Cite this Article:**Pradeep Maan, Amit Sehgal, Archana Malik, Number of irreducible factors and degree in divisor graph of Zp[x,n], J. Math. Comput. Sci., 11 (2021), 1364-1380 Copyright © 2021 Pradeep Maan, Amit Sehgal, Archana Malik. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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