Some properties of graphs associated with certain finite rings / Wan Muhammad Afif Wan Ruzali

Wan Muhammad Afif, Wan Ruzali (2019) Some properties of graphs associated with certain finite rings / Wan Muhammad Afif Wan Ruzali. Masters thesis, University of Malaya.

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Abstract

In this thesis, we investigate the properties of some graphs associated with finite rings, namely the total graphs, the unit graphs, and the directed graphs (also called digraphs). We start by describing the structure of the total graphs and unit graphs of some rings, particularly those of order p, p2 and pq where p and q are distinct primes. This involves computing the degrees of vertices and the number of connected components in those graphs. We also obtain explicit formulae for determining the number of sources or non-sources in the digraphs of certain rings. of special importance is the result related to the digraph associated with the ring of integers modulo pn, which is then used to extend the result on the number of sources to include the digraph associated with the more general ring Zn. Some graph parameters concerning the digraph associated with the ring I = ha, b : pa = pb = 0, a2 = b, ab = 0i of order p2 are also obtained, including the number of non-sources, the number of connected components, and the number of vertices in each connected component.

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