On the Defining Spectrum of k-Regular Graphs with k –1 Colors

In a given graph G = (V;E), a set of vertices S with an assignment of colors to them is said to be a defining set of the vertex coloring of G, if there exists a unique extension of the colors of S to C≥X(G) coloring of the vertices of G. A defining set with minimum cardinality is called a minimum defining set and its cardinality is the defining number, denoted by d(G; c). If F is a family of graphs thencSpec,(F)={d|…G,G element of F,d(G,C)=d} Here we study the cases where F is the family of k-regular (connected and disconnected) graphs on n vertices and c = k-1. Also the Speck-1(F) defining spectrum of all k-regular (connected and disconnected) graph on n vertices are verified for k = 3; 4 and 5.