Geodetic number and domination number of Γ(R(+)M)

This article investigates two important parameters of graph theory, the geodetic number and the domination number, in terms of zero divisor graphs for idealization rings R(+)M: the study offers a systematic analysis of these parameters, considering cases where R is an entirely integral domain and, in contrast, it is not. For the geodetic number, we give explicit formulas in several cases, including instances such as when R is an entirely consistent domain but also to certain constructions of Z(N)(+)Z(M) where N and M are related as prime powers. Similarly, for the domination number, we obtain exact values under various algebraic conditions. These results reveal interesting ties between the algebraic structure of the idealization ring and its associated zero divisor graph's geometric properties.