| Abstract: | For a positive integer n and a finite group G, let the symbols e(G, n) and E(G, n) denote, respectively, the smallest and the greatest number of lines among all n-point graphs with automorphism group G. We say that the Intermediate Value Theorem (IVT) holds for G and n, if for each e satisfying e(G, n)≤e≤E(G, n), there exists an n-point graph with group G and e lines. The main result of this paper states that for every group G the IVT holds for all sufficiently large n. We also prove that the IVT holds for the identity group and all n, and exhibit examples of groups for which the IVT fails to hold for small values of n. |
