The center of a graph G, denoted C(G), is the set of vertices of minimum eccentricity. A graph G is called pancentral if given any vertex, v∈V(G), there exists a spanning tree T of G such that v∈C(T). A pancentral graph G is called pan-unicentral if given any vertex, v∈V(G), there exists a spanning tree, T, of G such that the center, C(T) = {v}. A pancentral graph is called pan-bicentral if given any pair of adjacent vertices, u, v∈V(G), there exists a spanning tree T of G such that C(T) = {u,v} (Buckley, Lewinter, A friendly introduction to graph theory. Prentice-Hall, Upper Saddle River, 2003; Aulicino, Lewinter, Pancentral graphs. Cong Num 150:69–72, 2001; Aulicino, Lewinter, Almost all meshes are pancentral. GTN NY XXXIX:38–40, 2000). G is an F-graph if (a) |C(G)| ≥ 2, and (b) if x, y∈C(G), then d(x,y) = rad(G). An L-graph has the property that every diametral path of G contains at least one center vertex. Various properties are presented in this expository paper with the purpose of stimulating more research on these classes of graphs.