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We examine the problem of embedding a graph H as the center of a supergraph G, and we consider what properties one can restrict G to have. Letting A(H) denote the smallest difference I V(G)I -I V(H)I over graphs G having center isomorphic to H it is demonstrated that A(H)54 for all H, and for 0 5 i 5 4 we characterize the class of trees T withA(T) = i. For n 2 2 and any graph H, we demonstrate a graph G with point and edge connectivity equal to n , with chromatic number x(G) = n 4-x(H), and whose center is isomorphic to H. Finally, if I V(H) I 2 9 and k 2 I V(H) I + 1, then for n sufficiently large (with n even when k is odd) we can construct a k-regular graph on n vertices whose center is isomorphic to H.

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Abstract. Following Erd6s, Harary, and Tutte, the euclidean dimension of a graph G is the minimum n such that G can be embedded in euclidean n-space R" so that each edge of G has length 1. We present constructive proofs which give the euclidean dimension of a wheel and of a complete tripartite graph. We also define the generalized wheel W.,,. as the join/(,. + C. and determine the euclidean dimension of all generalized wheels.

The distance between a pair of vertices u, u in a graph G is the length of a shortest path joining u and u. The diameter diam(G) of G is the maximum distance between all pairs of vertices in G. A spanning tree Tof G is diameter preserving if diam(T) = diam(G). In this note, we characterize graphs that have diameter-preserving spanning trees.The distance d,(u, u ) between a pair of vertices u and u in a graph G is the length of a shortest path joining u and u. If u and u are in different components of G , then d,(u, u) = x . The eccentricity ec(u) of vertex u in G is its distance to a farthest vertex. When the graph in question is clear, we simply write d (u, u ) and e(u), omitting the subscript. If u is a vertex such that d(u, u ) = e(u), then u is called an eccentric vertex of u . The diameter, diam(G), and the radius, r(G), are the maximum and minimum eccentricities, respectively. The center C ( G ) is the set of vertices with minimum eccentricity and the periphery is the set of vertices with maximum eccentricity. Thus, a center vertex has e(u) = r ( G ) , while a peripheral vertex has e(u) = diam(G).Lesniak [2] characterized sequences of integers that are realizable as the sequence of eccentricities of a graph. More recently, Nandakumar [3] characterized graphs G with a spanning tree T that preserves the eccentricities, i.e., eT(u) = ec(u) for each vertex u E G as follows: