| Let G= (V, E) be a graph, A function f:V→{-1,0,1} defined on the the vertices of G is a minus dominating function, if the sum of its function values over any closed neighborhood is at least one. That is, for every vertex v∈V, A minus dominating function f is minimal, if there does not exist any minus dominating function g:V→{-1,0,1},f≠g, for which g(v)≤f(v) for every vertex v∈V. The minus domination number of a graph G, denotedγ- (G) is the minimum weight of all the minus dominating functions of G; The upper minus domination number of a graph G, denotedΓ-(G), equals the maximum weight of all the minimal minus dominating functions of G. That isγ-(G)= min{ω(f)|f is a minus dominating function on G} andΓ-(G)= max{ω(f)|f is a minimal minus dominating function on G}.Let G= (V, E) be a graph, A function∫:E→{-1,0,1} defined on the the edges of G is a minus edge domination function, if the sum of its function values over any closed neighborhood is at least one. That is for every e∈E. The minus edge domination number of a graph G, denotedγ'm(G) is the minimal weight of all the minus edge dominating functions of G; The upper minus edge domination number of a graph G, denotedΓ'm(G), is the maximum weight of all the minimal minus edge domination functions of G. That isγ'm(G)=min{ω(f)|f is a minus edge domination function on G} and Γ'm(G)=max{ω(f)|f is a minimal minus edge domination function on G}.By considering the structural properties of graphs,this thesis studies the upper mi-nus domination number of regular graphs.The main results obtained in this dissertation may be summarized as follows:(1) If G is a three regular graph of order n,thenΓ-(G)≤5/8n,the result is sharp and there exist a class of graphs,for whichΓ-(G):5/8n;If G is a four regular graph of order n,thenΓ-(G)≤7/(10)n；If G is a five regular graph of order n,thenΓ-(G)≤3/4n；(2)If G is a k-regular graph of order n,thenΓ-(G)≤(2k-1)/(2(k+1))n；(3)If G is a three regular graph of size m,thenΓ'm(G)≤(2m)/3. |
