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 a \(c \geq \chi(G)\) 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 then \( Spec_{c}(F)=\{d| \exists G, G \epsilon F, d(G,C)=d \} \). 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 \(Spec_{k-1}(F)\) defining spectrum of all \(k-\)regular (connected and disconnected) graph on n vertices are verified for \(k = 3, 4\) and \(5\).