Journal of Prime Research in Mathematics

# On the defining spectrum of $$k$$-regular graphs with $$k –1$$ colors

Doostali Mojdeh
Department of Mathematics, University of Mazandaran, Babolsar, Iran.

$$^{1}$$Corresponding Author: dmojdeh@umz.ac.ir

### Abstract

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 $$5$$.

#### Keywords:

Regular graphs; coloring; defining spectrum.