Abstract
The line graph for the complement of the zero divisor graph for the ring of Gaussian integers modulo n is studied. The diameter, the radius and degree of each vertex are determined. Complete characterization of Hamiltonian, Eulerian, planer, regular, locally and locally connected is given. The chromatic number when is a power of a prime is computed. Further properties for and are also discussed.
Highlights
3) a bi, a bi where a2 b2 = p, p is a prime integer and p 1 mod 4
The chromatic number when n is a power of a prime is computed
Throughout this paper, p and pi denote prime integers which are congruent to 1 modulo 4, while q and and qi denote prime integers which are congruent to 3 modulo 4
Summary
The zero divisor graph for the ring of Gaussian integers modulo n is studied in [1] and [2], the complement of this graph is discussed in [3]. While the line graph of the zero divisor graph for the ring of Gaussian integers modulo n is investigated in [4]. Q2 [i] is a complete graph whose complement is totally disconnected and its line graph is K0. Is derived, the degree of its complement as well as its line graph could be found. For a finite ring R , the line graph L R of a connected graph. The following theorem characterizes graphs G whose line graph L G is planer. Theorem 2.3 The graph L n[i] is planer if and only is n = 5.
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