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In this paper, we characterize the hyperbolic product graphs for the Cartesian sum : is always hyperbolic, unless either or is the trivial graph (the graph with a single vertex); if or is the trivial graph, then is hyperbolic if and only if or is hyperbolic, respectively. Besides, we characterize the Cartesian sums with hyperbolicity constant for every value of t. Furthermore, we obtain the sharp inequalities for every non-trivial graphs . In addition, we obtain simple formulas for the hyperbolicity constant of the Cartesian sum of many graphs. Finally, we prove the inequalities for the complement graph of for every with >= 3.
cartesian sum of graphs; geodesics; gromov hyperbolicity; complement of graphs