Lower connectivities of regular graphs with small diameter Original Research Article

Krishnamoorthy et al. [Minimum order graphs with specified diameter, connectivity and regularity, Networks 19 (1989) 25–46.] showed that a image-regular graph with diameter D at most 3 has (vertex-)connectivity image at least 2, and if image then the connectivity is at least image. Likewise, Soneoka et al [Sufficient conditions for maximally connected dense graphs, Discrete Math. 63 (1) (1987) 53–66] proved that a graph with diameter image has maximum connectivity image. In this work we generalize and improve these results for image-regular graphs. More precisely we prove that if image then image, and if image then image. Furthermore, we prove for g even that if image then image, and for bipartite image-regular graphs we obtain that if image then image, and if image then image. We establish similar bounds for the edge connectivity and present some examples showing that these results are best possible, at least for particular values of the girth and the regularity image.