Graphs with specified radius and diameter

The radius and diameter of a graph are known to satisfy the relation rad G < diam C 5 2 rad G. We show that this is the only restriction on these parameters and construl*t all nonisomorphic graphs of minimal order having a specified radius and diameter. In a graph G the diameter is diam G = ma,\_inax, d(tc, u) and the radius is rad G = min, max, d(tl, u), the min and max being taken over all points u and u of G. It is readily established that rad G 5 diam G 5 2 rad G. In fact this is the only restriction. We sho?v that for all positive integers m and yt satisfying m <_ n <: 2m, there exist graphs of radius m and diameter yt. We determine the minimum order of such a gray& and exhibit those of minimum order. For notation and terminology see [ 11 D There are essentially two cases to consider, according as y1 < 2~7t-2 or n 2 2m-1. In the latter case, the result is almost self-evident. For ti = 2m1 or n = 2m, a path of length y1 is of diameter n and radius m. Since every graph of diameter YI contains a path of length YE, n + 1 is the minimum order and the path of length y1 is the unique example of minimum order. The other case is a bit more complex. Theorem. For all positive integers m a,zd n satisfying m < n < 2m 2 there exist graphs of radius m and diameter n. The mimkwm ordw o.f such a graph is n + m. There are wactlv [$(n-m)] t 1 non-isomcbrphtc graphs oj’order n t m, radius in and diameter n. They are charactwized as being the minimal graphs which contain a geodesic path oj’lertgth 12 * Supporte.1 ir: part bv AFOSR grant No. 72-2163. 72 P.A. Ostrand, Graphs w&h specified radius and diameter