Minimum order graphs with specified diameter, connectivity, and regularity

Relationships among graph invariants such as the number of vertices, diameter, connectivity, maximum and minimum degrees, and regularity are being studied recently, motivated by their usefulness in the design of fault-tolerant and low-cost communication and interconnection networks. A graph is called a (d,c,r) graph if it has diameter d, connectivity c, and regularity r. The minimum number of vertices in (d, 1,3), (d,2,3), (d,3,3), and (d,c,c) graphs have been reported in the literature. In this paper, the minimum number of vertices in a (d,c,r) graph with r > c is determined, thereby exhausting all the possible choices of values for d, c, and r. Our proof is constructive and hence we get a collection of optimal (d,c,r) graphs.

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