Some Characteristics of the Prime Graph of Integer Modulo Groups
Abstract
The notion of the prime graph of a ring R was first introduced by Bhavanari, Kuncham, and Dasari in 2010. The prime graph of a ring R, denoted by PG(R) is a graph whose vertices are all elements of the ring, where two distinct vertices x and y are adjacent if and only if xRy = 0 or yRx = 0. In this paper, we study the forms and properties of the prime graph of integer modulo group, and some examples of the number of its spanning trees. In this paper, it is found that for all n, the maximum degree of vertices of PG(Z_n) is n-1 and the minimum degree of its vertices is 1. Then, we show that for all n, PG(Z_n) is neither a Hamiltonian graph nor an Eulerian graph. We also found some examples of the number of its spanning trees.
Keywords: prime graph; spanning trees; Hamiltonian graph.
Abstrak
Konsep mengenai graf prima dari suatu gelanggang R pertama kali diperkenalkan oleh Bhavanari, Kuncham, dan Dasari pada tahun 2010. Graf prima dari suatu gelanggang R, yang dinotasikan dengan PG(R), adalah suatu graf yang simpul-simpulnya merupakan semua elemen dari gelanggang tersebut dengan dua buah simpul x dan y yang berbeda akan bertetangga jika dan hanya jika xRy = 0 atau yRx = 0. Di dalam penelitian ini, dikaji mengenai bentuk-bentuk dan sifat-sifat dari PG(Z_n), dan beberapa contoh dari banyak pohon pembangunnya. Pada penelitian ini, ditemukan hasil bahwa untuk setiap n, derajat maksimal dari simpul-simpul di PG(Z_n) adalah n-1 dan derajat minimum dari simpul-simpulnya adalah 1. Hasil selanjutnya yaitu, untuk setiap n, PG(Z_n) bukan merupakan suatu graf Hamiltonian atau graf Eulerian. Ditemukan juga beberapa contoh dari banyaknya pohon pembangun dari PG(Z_n).
Kata Kunci: graf prima; pohon pembangun; graf Hamiltonian.
2020MSC: 05E16, 05C90, 20C05
Keywords
References
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DOI: 10.15408/inprime.v5i1.29014
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