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.