Algebraic Structures and Combinatorial Properties of Unit Graphs in Rings of Integer Modulo with Specific Orders

Sahin Two Lestari; Putu Kartika Dewi; I Gede Adhitya Wisnu Wardhana; I Nengah Suparta

doi:10.29303/emj.v7i2.235

Authors

Sahin Two Lestari Universitas Mataram
Putu Kartika Dewi Universitas Pendidikan Ganesha
I Gede Adhitya Wisnu Wardhana Universitas Mataram
I Nengah Suparta Universitas Pendidikan Ganesha

DOI:

https://doi.org/10.29303/emj.v7i2.235

Keywords:

numerical invariant, unit graph, ring of integer modulo

Abstract

Unit graph is the intersection of graph theory and algebraic structure, which can be seen from the unit graph representing the ring modulo n in graph form. Let R be a ring with nonzero identity. The unit graph of R, denoted by G(R), has its set of vertices equal to the set of all elements of R; distinct vertices x and y are adjacent if and only if x + y is a unit of R. In this study, the unit graph, which is in the ring of integers modulo n, denoted by G(Zn). It turns out when n is 2^k, G(Zn) forms a complete bipartite graph for k∈N, whereas when n is prime, G(Zn) forms a complete (n+1)/2-partites graph. Additionally, the numerical invariants of the graph G(Zn), such as degree, chromatic number, clique number, radius, diameter, domination number, and independence number complement the characteristics of G(Zn) for further research.

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