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 minimum eccentricity is called the radius, r(G), of the graph and the maximum eccentricity is called the diameter, d(G), of the graph. The super-radial graph R*(G) based on G has the vertex set as in G and two vertices u and v are adjacent in R*(G) if the distance between them in G is greater than or equal to d(G) − r(G) + 1 in G. If G is disconnected, then two vertices are adjacent in R*(G) if they belong to different components. A graph G is said to be a super-radial graph if it is a super-radial graph R*(H) of some graph H. The main objective of this paper is to solve the graph equation R*(H) = G for a given graph G.
@article{bwmeta1.element.doi-10_7151_dmgt_1769,
author = {K.M. Kathiresan and G. Marimuthu and C. Parameswaran},
title = {Characterization Of Super-Radial Graphs},
journal = {Discussiones Mathematicae Graph Theory},
volume = {34},
year = {2014},
pages = {829-848},
zbl = {1303.05049},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_7151_dmgt_1769}
}
K.M. Kathiresan; G. Marimuthu; C. Parameswaran. Characterization Of Super-Radial Graphs. Discussiones Mathematicae Graph Theory, Tome 34 (2014) pp. 829-848. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_7151_dmgt_1769/
[1] J. Akiyama, K. Ando and D. Avis, Eccentric graphs, Discrete Math. 56 (1985) 1-6. doi:10.1016/0012-365X(85)90188-8
[2] R. Aravamuthan and B. Rajendran, Graph equations involving antipodal graphs, presented at the Seminar on Combinatorics and Applications held at ISI, Calcutta during 14-17 December (1982), 40-43.
[3] R. Aravamuthan and B. Rajendran, On antipodal graphs, Discrete Math. 49 (1984) 193-195. doi:10.1016/0012-365X(84)90117-1
[4] R. Aravamuthan and B. Rajendran, A note on antipodal graphs, Discrete Math. 58 (1986) 303-305. doi:10.1016/0012-365X(86)90148-2
[5] F. Buckley and F. Harary, Distance in Graphs (Addition-Wesley, Reading, 1990). | Zbl 0688.05017
[6] F. Buckley, The eccentric digraphs of a graph, Congr. Numer. 149 (2001) 65-76. | Zbl 0989.05034
[7] E. Prisner, Graph Dynamics (Longman, London, 1995).
[8] G. Johns and K. Sleno, Antipodal graphs and digraphs, Internat. J. Math. Soc. 16 (1993) 579-586. doi:10.1155/S0161171293000717 | Zbl 0783.05045
[9] G. Johns, A simple proof of the characterization of antipodal graphs, Discrete Math. 128 (1994) 399-400. doi:10.1016/0012-365X(94)90131-7
[10] Iqbalunnisa, T.N. Janakiraman and N. Srinivasan, On antipodal eccentric and supereccentric graph of a graph, J. Ramanujan Math. Soc. 4(2) (1989) 145-161. | Zbl 0688.05041
[11] J. Boland, F. Buckley and M. Miller, Eccentric digraphs, Discrete Math. 286 (2004) 25-29. doi:10.1016/j.disc.2003.11.041 | Zbl 1048.05034
[12] J. Gimbert, M. Miller, F. Ruskey and J. Ryan, Iterations of eccentric digraphs, Bull. | Zbl 1079.05032
Inst. Combin. Appl. 45 (2005) 41-50.
[13] J. Gimbert, N. Lopez, M. Miller and J. Ryan, Characterization of eccentric digraphs, Discrete Math. 306 (2006) 210-219. doi:10.1016/j.disc.2005.11.015 | Zbl 1083.05036
[14] KM. Kathiresan and G. Marimuthu, A study on radial graphs, Ars Combin. 96 (2010) 353-360. | Zbl 1249.05095
[15] KM. Kathiresan and G. Marimuthu, Further results on radial graphs, Discuss. Math. Graph Theory 30 (2010) 75-83. doi:10.7151/dmgt.1477 | Zbl 1215.05047
[16] KM. Kathiresan, G. Marimuthu and S. Arockiaraj, Dynamics of radial graphs, Bull. Inst. Combin. Appl. 57 (2009) 21-28. | Zbl 1214.05017
[17] KM. Kathiresan and R. Sumathi, Radial digraphs, Kragujevac J. Math. 34 (2010) 161-170. | Zbl 1265.05189
[18] KM. Kathiresan, S. Arockiaraj and C. Parameswaran, Characterization of supereccentric graphs, submitted. | Zbl 1221.05116
[19] M.I. Huilgol, S.A.S. Ulla and A.R. Sunilchandra, On eccentric digraphs of graphs, Appl. Math. 2 (2011) 705-710. doi:10.4236/am.2011.26093
[20] N. López, A generalization of digraph operators related to distance properties in digraphs, Bulletin of the ICA 60 (2010) 49-61. | Zbl 1223.05053
[21] R.R. Singleton, There is no irregular Moore graph, Amer. Math. Monthly 75 (1968) 42-43. doi:10.2307/2315106 | Zbl 0173.26303
[22] D.B. West, Introduction to Graph Theory (Prentice-Hall of India, New Delhi, 2003).
[23] X. An and B. Wu, The Wiener index of the kth power of a graph, Appl. Math. Lett. 21 (2008) 436-440. doi:10.1016/j.aml.2007.03.025