J. Indones. Math. Soc.
Vol. 18, No. 2 (2012), pp. 113–118.
THE ECCENTRIC DIGRAPH OF THE CORONA OF
C
nWITH
K
m,
C
mOR
P
mSri Kuntari
1and Tri Atmojo Kusmayadi
21Department of Mathematics, Faculty of Mathematics and Natural Sciences
Sebelas Maret University [email protected]
2Department of Mathematics, Faculty of Mathematics and Natural Sciences
Sebelas Maret University [email protected]
Abstract. LetGbe a graph with a set of verticesV(G) and a set of edgesE(G). The distance from vertexu to vertexv inG, denoted byd(u, v), is the length of the shortest path from vertexuto v. The eccentricity of vertex u in graphG is the maximum distance from vertex u to any other vertices in G, denoted bye(u). Vertex v is an eccentric vertex from u if d(u, v) =e(u). The eccentric digraph ED(G) of a graphGis a graph that has the same set of vertices as G, and there is an arc (directed edge) joining vertexutovifvis an eccentric vertex fromu. In this paper, we answer the open problem proposed by Boland and Miller [1] to find the eccentric digraph of various classes of graphs. In particular, we determine the eccentric digraph of the corona ofCn withKm, Cmand Pm, withCn, KmorPm
are cycle, complete graph and path, respectively.
Key words: Eccentricity, eccentric digraph, corona graph.
2000 Mathematics Subject Classification: 05C99.
Abstrak. MisalGadalah suatu graf dengan himpunan titikV(G) dan himpunan sisiE(G) . Jarak dari titiku ke titikv diG, dinotasikand(u, v), adalah panjang dari lintasan terpanjang dari titikukev. Eksentrisitas titikudalam grafGadalah jarak maksimum dari titiku ke sebarang titik yang lain di G, dinotasikane(u). Titikv adalah suatu titik eksentrik dari u jikad(u, v) = e(u). Digraf eksentrik ED(G) dari suatu grafGadalah suatu graf yang mempunyai himpunan titik yang sama dengan himpunan titikG, dan terdapat suatu busur (garis berarah) yang menghubungkan titiku kev jika v adalah suatu titik eksentrik dari u. Boland dan Miller [1] memperkenalkan digraf eksentrik dari suatu digraf. Mereka juga mengusulkan suatu masalah untuk menemukan digraf ekesentrik dari bermacam kelas dari graf. Dalam makalah ini diselidiki eksentrik digraf dari graf koronaCn
denganKm, CmatauPm, denganCn, KmdanPmmasing-masing adalah graf siklik,
graf lengkap dan lintasan.
Kata kunci: Eksentrisitas, digraf eksentrik, graf korona.
1. Introduction
Most of the notations and terminologies follow that of Gallian [5], Chartrand and Oellermann [3]. Let G be a graph with a set of vertices V(G) and a set of edgesE(G). Thedistance from vertexuto vertexvinG, denoted byd(u, v), is the length of the shortestpath from vertexuto v. If there is no apath joining vertex
uand vertex v, thend(u, v) =∞. The eccentricity of vertex uin graph Gis the maximumdistance from vertexuto any other vertices inG, denoted bye(u), and soe(u) =max{d(u, v)|v∈V(G)}. Radiusof a graphG, denoted byrad(G), is the minimum eccentricity of every vertex in G. Thediameter of a graph G, denoted bydiam(G), is the maximumeccentricity of every vertex in G. Ife(u) =rad(G), then vertex uis called central vertex. Center of a graphG, denoted bycen(G), is an induced subgraph formed from central vertices of G. Vertex v is an eccentric
vertex from uif d(u, v) = e(u). The eccentric digraph ED(G) of a graphG is a graph having the same set of vertices asG,V(ED(G)) =V(G), and there is anarc
(directed edge) joining vertexutov ifv is aneccentric vertex fromu. Anarcof a digraphDjoining vertexutovand vertexvtouis called asymmetric arc. Further, Fred Buckley [2] concluded that almost in every graphG, itseccentric digraph is
ED(G) = G∗, where G∗ is a complement of G which is every edge replaced by a symmetric arc. One of the topics in graph theory is to determine theeccentric
digraph of a given graph. Theeccentric digraphof a graph was initially introduced by Fred Buckley (Boland and Miller [1]). The class of graph considered here is
corona of two graphsG1and G2. According to Harary [6], thecorona G1⊙G2of two graphs G1 andG2was defined as the graph Gobtained by taking one copy of
G1(which has n vertices) and n copies ofG2 and then joining theith vertex ofG1
to every vertex in theith copy ofG2. Some authors have investigated the problem of finding theeccentric digraph. For example, Boland and Miller [1] determined the
The Eccentric Digraph 115
to find the eccentric digraph of various classes of graphs. Some results related to this open problem can be found in [7, 8]. In this paper, we also answer the open problem proposed by Boland and Miller [1]. In particular, we determine the
eccentric digraphof thecorona of a cycle of n verticesCn with the classes of graphs Km, CmorPm, whereCm, KmandPmare cycle, complete graph and path with m
vertices, respectively.
2. Main Results
Given two graphs Cn and M, where M is either a cycle Cm, a complete
graph Km or a pathPm. The set of vertices of them areV(Cn) ={v1, v2,· · ·, vn}
and V(M) = {u1, u2,· · · , um}, respectively. The edges set of Cn is E(Cn) =
{v1v2, v2v3,· · ·, vn−1vn, vnv1}. The set of edges ofM areE(M) ={u1u2, u2u3,· · · ,
um−1um, umu1} for M = Cm, E(M) = {uiuj : i, j = 1,2,· · ·, m} for M =Km
and E(M) ={u1u2, u2u3,· · ·, um−1um} forM =Pm. The corona of Cn withM,
denoted byCn⊙M is a graph with V(Cn⊙M) =V(Cn∪∪vi∈V(Cn)V(Mi) and
E(Cn⊙M) = E(Cn)∪
∪
vi∈CnE(Mi)∪ {(vi, ui : vi ∈ V(Cn), ui ∈ V(Mi)} with
Mi=M.
The materials of this research are mostly from the papers related to the eccentric digraph. The following result is the eccentric digraphs of the corona
Cn⊙M graph. There are two cases the eccentric digraphs of the coronaCn⊙M
graph to consider based on the values ofn.
Theorem 2.1. Let Cn⊙M be the corona graph ofCn withM, whereM is either
one ofCm,KmorPm. Ifnis even then the eccentric digraph ED(Cn⊙M)is the
graph Gas depicted in Figure 1.
Proof. It is easy to check that the eccentricity of vertexviis n
2+ 1 and the
eccen-tricity of vertexuij is n2+ 2,i= 1,2,· · ·, nandj= 1,2,· · · , m. The eccentricities
of all vertices are used to determine the eccentric vertex of all vertices of the corona graph Cn⊙M. The arcs can be obtained by joining every vertex to its eccentric
vertex of the corona graphCn⊙M. Table 1 shows the eccentric vertices and arcs
of the corona graphCn⊙M.
From Table 1, the arc of the corona graphCn⊙M vertex is adjacent to its
eccentric vertices. The arcsuiju(n
2+i(mod n))jare symmetric and arcsviu( n
2+i(mod n))j
are not symmetric fori= 1,2,· · ·, n, j= 1,2,· · · , m. Therefore the eccentric digraph of the corona graph Cn⊙M with n even can be formed into n2Km,m∪ n(K1∨Km) withV(K1) ={vi},Km=V(Mn
2+i(mod n)) fori= 1,2,· · ·, n.
Theorem 2.2. Let Cn⊙M be the corona graph ofCn withM, whereM is either
one ofCm,Km orPm. If nis odd then the eccentric digraph ED(Cn⊙M)is the
Table 1. The eccentric vertices and arc ofCn⊙M vertex, whereneven
vertex of Cn⊙M eccentric vertices arc
v1 u(n
Figure 1. The eccentric digraph ofCn⊙M withneven
Proof. By observation, we obtain the eccentricity of vertexvi is n+1
2 and
eccen-tricity of vertexuij is n+12 + 1,i= 1,2,· · · , nandj= 1,2,· · · , m. The eccentricity
of all vertices are used to determine the eccentric vertex of all vertices of the corona graph Cn⊙M. The arcs can be obtained by joining every vertex to its eccentric
vertex of the corona graphCn⊙M. Table 2 shows the eccentric vertices and arcs
of the corona graphCn⊙M.
From Table 2, the arc of the corona graphCn⊙M vertex is adjacent to its
eccentric vertices. The symmetric arcs areuiju(n+1
The Eccentric Digraph 117
Table 2. The eccentric vertices and arc ofCn⊙M vertex, wherenodd
vertex ofCn⊙M eccentric vertices arc
v1 un+1
Figure 2. The eccentric digraph ofCn⊙M withnodd
References
[1] Boland, J. and Miller, M.,”The Eccentric Digraph of a Digraph”,Proceeding of AWOCA’01 Lembang-Bandung, Indonesia,(2001), 66-70.
[2] Buckley, F., ”The Eccentric Digraph of a Graph”,Congressus Numerantium149(2001),65-76.
[3] Chartrand, G. and Oellermann, O. R.,Applied and Algorithmic Graph Theory, International Series in Pure and Applied Mathematics, McGraw-Hill Inc, California, 1993
[5] Gallian, J. A. ”Dynamic Survey of Graph Labeling”,The Electronic Journal of Combina-torics,16,(2009), 1-219.
[6] Harary, F.,Graph Theory, Addison-Wesley Publishing Company.Inc.,Reading, MA, 1972. [7] Kusmayadi, T.A. and M. Rivai. ”The Eccentric Digraph of an Umbrella Graph”,Proceeding of
INDOMS International Conference on Mathematics and Its Applications (IICMA), Gadjah Mada University Yogyakarta, Indonesia,ISBN 978-602-96426-0-5,(2009), 0627-0638 [8] Kusmayadi, T.A. and M. Rivai. ”The Eccentric Digraph of a Double Cones Graph”,
