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In any network, the diameter is often taken as a measure of efficiency, as it measures the maximum communication delay. The fact that addition or deletion of edges changes the diameter gives rise to the concept of diameter variability. Its study becomes important as it determines the communication efficiency when an addition or deletion of a link occurs in a network. Two other important graph parameters considered as a measure of reliability and efficiency of networks are wide diameter and fault diameter. In this paper, we study these diameter notions of the Fibonacci cubes.

The diameter of a graph can be affected by the addition or deletion of edges. In this paper, we examine the Cartesian product of graphs whose diameter increases (decreases) by the deletion (addition) of a single edge. The problems of minimality and maximality of the Cartesian product of graphs with respect to its diameter are also solved. These problems are motivated by the fact that most of the interconnection networks are graph products and a good network must be hard to disrupt and the transmissions must remain connected even if some vertices or edges fail.