Geodetic spectra of graphs

Geodetic numbers of graphs and digraphs have been investigated in the literature recently. The main purpose of this paper is to study the geodetic spectrum of a graph. For any two vertices u and v in an oriented graph D, a u–v geodesic is a shortest directed path from u to v. Let I(u,v) denote the set of all vertices lying on a u–v geodesic. For a vertex subset A, let I(A) denote the union of all I(u,v) for u,v∈A. The geodetic number g(D) of an oriented graph D is the minimum cardinality of a set A with I(A)=V(D). The (strong) geodetic spectrum of a graph G is the set of geodetic numbers of all (strongly connected) orientations of G. In this paper, we determine geodetic spectra and strong geodetic spectra of several classes of graphs. A conjecture and two problems given by Chartrand and Zhang are dealt with.