Dynamics of Boundary Graphs

In a graph G, the distance d(u,v) between a pair of vertices u and v is the length of a shortest path joining them. A vertex v is a boundary vertex of a vertex u if ) , ( ) , ( v u d w u d  for all ). (v N w The boundary graph B(G) based on a connected graph G is a simple graph which has the vertex set as in G. Two vertices u and v are adjacent in B(G) if either u is a boundary of v or v is a boundary of u. If G is disconnected, then each vertex in a component is adjacent to all other vertices in the other components and is adjacent to all of its boundary vertices within the component. Given a positive integer m, the m iterated boundary graph of G is defined as )). ( ( ) ( 1 G B B G B m m   A graph G is periodic if G G B  ) ( for some m. A graph G is said to be an eventually periodic graph if there exist positive integers m and k >0 such that . ), ( ) ( k i G B G B i i m     We give the necessary and sufficient condition for a graph to be eventually periodic.

). (v N w  The boundary graph B(G) based on a connected graph G is a simple graph which has the vertex set as in G. Two vertices u and v are adjacent in B(G) if either u is a boundary of v or v is a boundary of u.If G is disconnected, then each vertex in a component is adjacent to all other vertices in the other components and is adjacent to all of its boundary vertices within the component.Given a positive integer m, the m th iterated boundary graph of G is defined as for some m.A graph G is said to be an eventually periodic graph if there exist positive integers m and k >0 such that .), ( )

Introduction and Definitions
The graphs considered here are nontrivial and simple.For other graph theoretic notation and terminology, we follow [1].In a graph G, the distance d(u,v) between a pair of vertices u and v is the length of a shortest path joining them.The eccentricity e(u) of a vertex u is the distance to a vertex farthest from u.The radius r(G) of G is defined as A vertex v of G is called an eccentric vertex of G if it is an eccentric vertex of some vertex of G.The eccentric graph based on G is denoted by G e , whose vertex set is V(G) and two vertices u and v are adjacent in G e if and only if d(u,v)= min{e(u),e(v)}.
Gimbert et al. [3] studied the iterations of eccentric digraphs.The eccentric digraph of a digraph G, denoted by ED(G), is the digraph on the same vertex set as in G but with an arc from a vertex u to a vertex v in ED(G) if and only if v is an eccentric vertex of u in G. Given a positive integer k, the k th iterated eccentric digraph of G is written as where ED 0 (G) = G.For every digraph G, there exists smallest integer p ' >0 and where  denotes graph isomorphism.We call p ' , the iso-period of G and t ' , the iso-tail of G; these quantities are denoted by p ' (G) and t ' (G), respectively.
Kathiresan and Marimuthu [4] introduced a new type of graph called radial graph.Two vertices of a graph G are said to be radial to each other if the distance between them is equal to the radius of the graph.Two vertices of graph G are said to be radial to each other if the distance between them is equal to the radius of the graph.The radial graph of a graph G denoted by R(G) has the vertex as in G and two vertices are adjacent in R(G) if and only if they are radial in G.
for some graph H.In [5] Kathiresan et al. studied the properties of iteration of radial graphs.Given a positive integer m, the m th iterated radial graph of G is defined as for some m.If p is the least positive integer with this property, then G is called a periodic graph with iso-period p.When p=1, G is called as a fixed graph.A graph G is said to be eventually periodic if there exist positive integers m and k>0, such that .), ( ) If p and k are the least positive integers with this property, then G is eventually periodic with iso-period p and iso-tail k.
Based on the concept of radial graphs, Marimuthu and Sivanandha Saraswathy [6] introduced the concept of boundary graphs.A vertex v is a boundary vertex of a vertex u The boundary graph B(G) based on a connected graph G is a simple graph which has the vertex set as in G. Two vertices u and v are adjacent in B(G) if either u is a boundary of v or v is a boundary of u.If G is disconnected, then each vertex in a component is adjacent to all the vertices in the other components and is adjacent to all of its boundary vertices within the component.A graph G is called a boundary graph if there exists a graph H such that B(H) = G. we defined the Motivated by the work of J. Gimbert et al., [2,3] and KM.Kathiresan et al., [5],We study here an iterated version of a distance dependent mapping.Given a positive integer m, the m th iterated boundary graph of G is defined as for some m.If p is the least positive integer with this property, then G is called a periodic graph with iso-period p.When p = 1, G is called as a fixed graph.Definition 1.2: A graph G is said to be eventually periodic if there exist positive integers m and k>0, such that .), ( ) If p and k are the least positive integers with this property, then G is called an eventually periodic graph with iso-period p and iso-tail k .
Figs. 1, 2 and 3 illustrate these definitions showing boundary graph of G and its iterated boundary graphs.F denote the set of all disconnected graphs.It is well known that d(G)  4 implies that 2. ) G d( 

Previous results
The following theorems are appeared in [6].
for any two adjacent vertices u and v of G.

Theorem 2.3 [6]:
Let G be a graph.Then B(G) = G if and only if the following conditions hold.
(i) G has no complete vertex .
for any two adjacent vertices u and v of G.
for any two non-adjacent vertices u and v of G , where k= d (u,v)+1.

Theorem 2.4 [6]:
If G has at least one isolated vertex, then G is not a boundary graph.
for any two adjacent vertices u and v of G.
for any two non-adjacent vertices u and v of G , where k= d (u,v)+1 then, G is a boundary graph.

Main Results
Proposition 3.1: Every graph is either periodic or eventually periodic.

Proof. Consider the set
. Thus, there exist non-negative integer k and positive integer m such that Proposition 3.2: Let C n be any cycle.Then C n is periodic with iso-period 1 if it is odd and eventually periodic with iso-period 1 if it is even.
Hence C n is periodic with iso-period 1.

Case(ii)
If n is even, Let us find some graphs of order n which is either periodic or eventually periodic.Hence by Theorem2.1,P n is eventually periodic with iso-period 1.
Lemma 3.9: A graph 12 F G  is eventually periodic with iso-period 1 if and only if either for any two adjacent vertices u and v of G.

Theorem 2 . 5 [ 6 ]: Let 4 F
G  without isolated vertices.If G without complete vertices has the following properties graph with each component K 2 .By the definition of B(G), graph.Hence by Theorem 2.1, C n is eventually periodic with iso-period 1.

Observation 3 . 3 :Observation 3 . 4 : 5 :Observation 3 . 6 : 1 . 3 . 8 :
C n +C n is a periodic graph for odd values of n,3  nwhose k(G)=0 and p(G)=2 where + denotes the usual addition of graphs.We also observed thatp(C m +C m ) = p(C m ) + p(C m ) where m = 2n+1, C 2m+1  C 2m+1 is a fixed graph whose k(G)=0 and p(G)=1 , 1  m where  denotes the Cartesian product of graphs.C 2m  C 2m is eventually periodic with k(G)=2, p(G)=1.Let us say that a class is periodic if every graph in the class is periodic.As we observed earlier C n +C n , complete graph , C 2m+1  C 2m+1 are periodic graphs.Observation 3.7: Every complete n-partite graph with |V i |  2 for each i th partition is eventually periodic with iso-period 1.Proof.Let G be a complete n-partite graph with |V i |  2 for each i th partition.Any two vertices v i and v j in G are adjacent in B(G) if and only if they are in the same partition.Therefore B(G) is a disconnected graph with each component complete.By the definition of boundary graph, B 2 (G) is complete.By Theorem 2.1, G is eventually periodic with isoperiod Proposition Every path P n , 3  n is eventually periodic with iso-period 1.