THE ECCENTRIC DIGRAPH OF FRIENDSHIP GRAPH, AND FIRECRACKER GRAPH.

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.
is a digraph having the vertex set V ( Dkn ) = {u , v1,1 , v1, 2 , K , v n.k −1 } and the arc set
The results show that (1) the eccentric digraph of friendship graph Fkn for k even,

⎫
k⎫ ⎧
k

k⎫
⎧
⎧
A( ED ( Fkn )) = ⎨uv i , j | i ∈ [1, n], j = ⎬ ∪ ⎨v k v k | i, j ∈ [1, n], i ≠ j ⎬ ∪ ⎨vi , j v s ,t | i, j ∈ [1, n], j ≠ , s ∈ [1, n], s ≠ i, t = ⎬ ,
,
,
i
j
2
2⎭
2⎭ ⎩ 2 2
⎩
⎩
⎭

(2) the eccentric digraph of friendship graph Fkn for k odd is a digraph having the vertex
set Vi = V ( K k −3,i ) = {vi ,1 , vi , 2 , K , v

and

the

arc

i,

k −3
2

,v
i,

set

k +3
2

, K , vi , k −1 } , Vi ' = V ( K 2,i ) = {v

graph Fn ,k

k −1
2

,v
i,

k +1
2

}

A( ED ( Fkn )) = {uα i | i ∈ Vi ' , i ∈ [1, n]} ∪

{α r α s | α r ∈ Vr ' , α s ∈ Vs ' , r ≠ s, r , s ∈ [1, n]} ∪ {β p β q | β p ∈ Vi , β q ∈ V j ' , i ≠ j ,

p = 1,2, K,

i,

i, j ∈ [1, n]} ,

k −3 k +3
k −1 k +1
,
,K, k − 1, q =
,
, (3) the eccentric digraph of firecracker
2
2
2
2
is a 4-partite F
nk − 2 k + 4 nk − 2 k + 4 , and (4) the eccentric digraph of firecracker
k − 2,k − 2,

2

graph Fn,k is a digraph 5-partite F

,

k − 2,k − 2,k ,

2
nk − 3 k + 4 nk − 3 k + 4
,
2
2

.