Center Smooth Sets and Center Smooth Numbers of Graphs

For any proper set S of V in a graph G ( V, E ), the S-eccentricity, e G, S ( υ ) (in short e S ( υ )) of a vertex υ in G is max x ∈ S ( d ( υ , x ) ) . The S-center of G is C S ( G ) = { υ ∈ V | e S ( x ) ≤ e S ( x )∀ x ∈ V } and S 1 -eccentricity, e G,S 1 ( v ) (in short e S 1 ( V )) of a vertex v in S is max x ∈ V = S ( d ( υ , x ) ) . S 1 -center of G is C S 1 ( G ) = { υ ∈ V | e S 1 ( x ) ≤ e S 1 ( x )∀ x ∈ V }. Then G is called a center-smooth graph if C S ( G ) = C S 1 ( G ) and the set S is defined to be a center-smooth set. We identify the center smooth sets of certain classes of graphs namely, K q,p , K p – q , K p , wheel graphs and lollipop graph and enumerate them for many of these graph classes. We also introduce the concept of center smooth number, which is defined as the number of distinct center smooth set of a graph G, and determine the center smooth number of some graph classes.