Geodetic Spectra Of Several Classes Of Graphs

Geodetic numbers of graphs and digraphs have been investigated in the litrature recently. The main purpose of these papers are to study the geodetic spectrum of a graph. For any two vertices u and v in 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 the u - v geodesies and the v - u geodesies. For a vertex set A, let I(A) denote the union of all I(u, v) for u,u ∈ A. The geodetic number g(D) of an oriented graph D is the minimun cardinary of a set A with I (A) = V(D). The geodetic spectrum of a graph G is the set of geodetic number of all orientation of G , which is denoted by S(G). In this paper, we determine the relations between the geodetic spectra of the graph G[(G₁,V₁), (G₂,v₂);v], which is obtained from the graph G₁ and G₂ by identifying a vertex V₁ of G₁ and a vertex V₂ of G₂ as a new vertex v, and the geodetic spectra of the graphs G₁,G₂ Combine the results of Gerard J. Chang et al.'s (Europ. J. Combinatorics. 25(2004), 303-391) and ours, we determine the geodetic spectra of several classes of graphs.