The convexity spectra of graphs

Let D   be a connected oriented graph. A set S⊆V(D)S⊆V(D) is convex in D   if, for every pair of vertices x,y∈Sx,y∈S, the vertex set of every x-yx-y geodesic (x-yx-y shortest dipath) and y-xy-x geodesic in D is contained in S. The convexity number  con(D)con(D) of a nontrivial oriented graph D is the maximum cardinality of a proper convex set of D. Let G   be a graph. We define that SC(G)={con(D):DSC(G)={con(D):D is an orientation of G}G} and SSC(G)={con(D):DSSC(G)={con(D):D is a strongly connected orientation of G}G}. In the paper, we show that, for any n⩾4n⩾4, 1⩽a⩽n-21⩽a⩽n-2, and a≠2a≠2, there exists a 2-connected graph G with n   vertices such that SC(G)=SSC(G)={a,n-1}SC(G)=SSC(G)={a,n-1} and there is no connected graph G   of order n⩾3n⩾3 with SSC(G)={n-1}SSC(G)={n-1}. Then, we determine that SC(K3)={1,2}SC(K3)={1,2}, SC(K4)={1,3}SC(K4)={1,3}, SSC(K3)=SSC(K4)={1}SSC(K3)=SSC(K4)={1}, SC(K5)={1,3,4}SC(K5)={1,3,4}, SC(K6)={1,3,4,5}SC(K6)={1,3,4,5}, SSC(K5)=SSC(K6)={1,3}SSC(K5)=SSC(K6)={1,3}, SC(Kn)={1,3,5,6,…,n-1}SC(Kn)={1,3,5,6,…,n-1}, SSC(Kn)={1,3,5,6,…,n-2}SSC(Kn)={1,3,5,6,…,n-2} for n⩾7n⩾7. Finally, we prove that, for any integers n, m, and k   with n⩾5,n+1⩽m⩽n2-1, 1⩽k⩽n-11⩽k⩽n-1, and k≠2,4k≠2,4, there exists a strongly connected oriented graph D with n vertices, m edges, and convexity number k.