# On the Relation of the Total Graph of a Ring and a Product of Graphs

## DOI:

https://doi.org/10.15642/mantik.2022.8.2.99-104## Keywords:

Group, Total graph, Isomorphism, Cartesian product## Abstract

The total graph of a ring R, denoted as T(Γ(R)), is defined to be a graph with vertex set V(T(Γ(R)))=R and two distinct vertices u,v∈V(T(Γ(R))) are adjacent if and only if u+v∈Z(R), where Z(R) is the zero divisor of R. The Cartesian product of two graphs G and H is a graph with the vertex set V(G×H)=V(G)×V(H) and two distinct vertices (u_1,v_1 ) and (u_2,v_2 ) are adjacent if and only if: 1) u_1=u_2 and v_1 v_2∈H; or 2) v_1=v_2 and u_1 u_2∈E(G). An isomorphism of graphs G dan H is a bijection ϕ:V(G)→V(H) such that u,v∈V(G) are adjacent if and only if f(u),f(v)∈V(H) are adjacent. This paper proved that T(Γ(Z_2p )) and P_2×K_p are isomorphic for every odd prime p.

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*Jurnal Matematika MANTIK*,

*8*(2), 99–104. https://doi.org/10.15642/mantik.2022.8.2.99-104

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Copyright (c) 2022 Mohammad Nafie Jauhari

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