On Regular-Stable Graphs

Abstract

We introduce graphs $G$ with at least one maximum independent set of vertices $I$, such that $\mathrm{\forall }v\in V(G)\setminus I$, the number of vertices in $N_G(v)\cap I$ is constant. When this number of vertices is equal to $\lambda$, we say that $I$ has the $\lambda$-property and that $G$ is $\lambda$-regular-stable. Furthermore, we extend the study of this property to the well-covered graphs (that is, graphs where all maximal independent sets of vertices have the same cardinality). In this study, we consider well-covered graphs for which all maximal independent sets of vertices have the $\lambda$-property, herein called well-covered $\lambda$-regular-stable graphs.