Extremal problems on saturation for the family of k-edge-connected graphs

Let F be a family of graphs. A graph G is F-saturated if G contains no member of F as a subgraph but G+e contains some member of .F whenever e is an element of E((G) over bar) The saturation number and extremal number of .F, denoted sat(n, F) and ex(n, .F) respectively, are the minimum and maximum numbers of edges among n-vertex .F-saturated graphs. For k is an element of N, let F-k and F-k' be the families of k-connected and k-edge-connected graphs, respectively. Wenger proved sat(n, F-k) = (k - 1)n - ((k)(2)); we prove sat(n, F-k') = (k - 1)(n - 1) -left perpendicular n/k+1 right perpendicular ((2) (k-1)). We also prove ex(n, F-k') = (k - 1)n - ((k)(2)) and characterize when equality holds. Finally, we give a lower bound on the spectral radius for F-k-saturated and F-k'-saturated graphs. (C) 2019 Elsevier B.V. All rights reserved.