SUMMARY
Tri Atmojo Kusmayadi, 2011. THE ECCENTRIC DIGRAPH OF FRIENDSHIP
GRAPH, AND FIRECRACKER GRAPH. Faculty of Mathematics and Natural Sciences,
Sebelas Maret University.
Let G be a graph with a set of vertices V(G) and a set of edges E(G). The distance
from vertex u to vertex v in G, denoted by d(u,v), is a length of the shortest path from
vertex u to v. The eccentricity of vertex u in graph G is the maximum distance from
vertex u to any other vertices in G, denoted by e(u). Vertex v is an eccentric vertex from u
if d(u,v)=e(u). The eccentric digraph ED(G) of a graph G is a graph that has the same set
of vertices as G, and there is an arc (directed edge) joining vertex u to v if v is an
eccentric vertex from u.
The purposes of this reserarch are to determine the eccentric digraphs of some
classes of graphs, in particular the friendship graphs, and the firecracker graphs.
The results show that (1)the eccentric digraph of friendship graph F for k even, kn
is a digraph having the vertex set ( )=
{
, 1,1, 1,2, , n.k−1}
n
k u v v v
D
V K and the arc set
∪ ⎭ ⎬ ⎫ ⎩
⎨
⎧ ∈ ≠
∪ ⎭ ⎬ ⎫ ⎩
⎨
⎧ ∈ =
= uv i n j k v v i j n i j
F ED
A k
j k i j
i n
k | , [1, ],
2 ], , 1 [ | ))
( (
2 , 2 , ,
⎭ ⎬ ⎫ ⎩
⎨
⎧ ∈ ≠ ∈ ≠ =
2 , ], , 1 [ , 2 ], , 1 [ , |
, ,
k t i s n s k j n j i v
vij st ,
(2) the eccentric digraph of friendship graph F for k odd is a digraph having the vertex kn
set ( ) { , , , , , , , 1}
2 3 , 2
3 , 2 , 1 , ,
3 − + −
− =
= k ik
i k i i
i i k
i V K v v v v v
V K K , ' ( ) { , }
2 1 , 2
1 , ,
2 = − +
= k
i k i i
i V K v v
V
and the arc set A(ED(Fkn))={u
α
i |i∈Vi',i∈[1,n]}∪∪ ∈
≠ ∈
∈ ', ', , , [1, ]} |
{αrαs αr Vr αs Vs r s r s n {βpβq |βp∈Vi,βq∈Vj', i≠ j, i, j∈[1,n]},
2 1 , 2
1 ,
1 , , 2
3 , 2
3 , , 2 ,
1 − + − = − +
= k k k q k k
p K K , (3) the eccentric digraph of firecracker
graph Fn,k is a 4-partite
2 4 2 , 2
4 2 , 2 ,
2 − − + − +
− k nk k nk k k
F , and (4) the eccentric digraph of firecracker
graph Fn,k is a digraph 5-partite
2 4 3 , 2
4 3 , , 2 ,
2 − − + − +
− k knk k nk k k
