Some Results On The Geodetic Number Of Graphs

The geodetic number of graphs is an important parameter revealing the structural character of graphs. The geodetic number of graphs originated from the convex set theory of geometry, topology and functional analysis and is the generalization and application of convex set theory in graph theory.In this thesis , we will consider the geodetic number of graphs and diagraphs , and research results of mine are mainly presented, the main content of the thesis involves the following four parts: (l)Some properties about the geodetic set of graphs and their orientations (2)Some bounds of the lower geodetic number g-(G) (3)Some endeavor for the problem g(G) ≤ g+(G) and some bounds of these two parameters (4)Several special graphs' geodetic spectrum under two particular oriental rules.Here is the main results of this thesis:1. Some relations between vertex-cut set,component and the geodetic set;2. For any connected graph G and any spanning tree T of G,g-(G) ≤ l(T),where l(T) is the number of leaves of T;3. g(G) ≤ g+(G) is true for chordal graphs or graphs with no 3-cycle or graphs with chromatic number x(G) ≤ 4 or x(G) ≥ n - 4;4. When n is large enough ,for almost every tournaments G of order n,G's geodetic numberOn the other hand when n ≥ 3 ,for every complete graph Kn ,the geodetic spectrum of Kn's strong connected orientations is...