In this paper, we prove that for any positive integers k, n with k ≥ 2, the graph P k n is a divisor graph if and only if n ≤ 2k + 2, where P k n is the k power of the path Pn. For powers of cycles we show that C n is a divisor graph when n ≤ 2k + 2, but is not a divisor graph when n ≥ 2k + bk2 c+ 3, where C k n is the k th power of the cycle Cn. Moreover, for odd n with 2k + 2 < n < 2k + bk2 c...

‎in this paper we give a characterization for all commutative‎ ‎rings with $1$ whose zero-divisor graphs are $c_4$-free.‎

return real } inline { if (x <= 0.0) { drunerr(205, x); errfail: } if is:null(b) then } inline { return C_double log(x); ) else { if !cnv:C_double(b) then runerr(102, b) b o d y { static double Iastbase = 0.0; static double divisor; if (b <= 1.0) { drunerr(205, b); errfail; } if (b != Iastbase) { divisor = log(b); Iastbase = b; } x = log(x) / divisor; return C_double x;

Let H be a Krull monoid with infinite class group such that each divisor class contains a prime divisor. We show that for every positive integer n, there exists a divisor closed submonoid S of H such that min∆(S) = n.

In this paper, we introduce a family of graphs which is a generalization of zero-divisor graphs and compute an upper-bound for the diameter of such graphs. We also investigate their cycles and cores

superelliptic curves, jacobians, cryptography, discrete logarithm problem In this paper we present an efficient, polynomial-time method to perform calculations in the divisor class group of a curve which has a single point on its normalization above infinity. In particular, we provide a unique representation of divisor classes and an algorithm for reducing a divisor on such a curve to its corre...