### On the Steiner antipodal number of graphs

#### Abstract

The Steiner *n*-antipodal graph of a graph *G* on *p* vertices, denoted by *S**A*_{n}(*G*), has the same vertex set as *G* and any *n*(2 ≤ *n* ≤ *p*) vertices are mutually adjacent in *S**A*_{n}(*G*) if and only if they are *n*-antipodal in *G*. When *G* is disconnected, any *n* vertices are mutually adjacent in *S**A*_{n}(*G*) if not all of them are in the same component. *S**A*_{n}(*G*) coincides with the antipodal graph *A*(*G*) when *n* = 2. The least positive integer *n* such that *S**A*_{n}(*G*) ≅ *H*, for a pair of graphs *G* and *H* on *p* vertices, is called the Steiner *A*-completion number of *G* over *H*. When *H* = *K*_{p}, the Steiner *A*-completion number of *G* over *H* is called the Steiner antipodal number of *G*. In this article, we obtain the Steiner antipodal number of some families of graphs and for any tree. For every positive integer *k*, there exists a tree having Steiner antipodal number *k* and there exists a unicyclic graph having Steiner antipodal number *k*. Also we show that the notion of the Steiner antipodal number of graphs is independent of the Steiner radial number, the domination number and the chromatic number of graphs.

#### Keywords

#### Full Text:

PDFDOI: http://dx.doi.org/10.5614/ejgta.2019.7.2.3

#### References

R.Aravamudhanand, B. Rajendran, Graph equations involving antipodal graphs, Presentedat the seminar on combinatorics and applications held at ISI, Culcutta during 14–17, December, (1982), 40–43.

R. Aravamudhanand, B. Rajendran, On antipodal graphs, Discrete Math.49 (1984), 193–195.

J.A. Bondy, U.S.R. Murty, Graph Theory with Applications, North Holland, New York, Amsterdam, Oxford, 1976.

F. Buckley and F. Harary, Distance in graphs, Addison-Wesley, Reading, 1990.

G. Chartrand, O.R. Oellermann, S. Tian and H.B. Zou, Steiner distance in graphs, Casopis

Pro Pestovani Matematiky 114 (4) (1989), 399–410.

F. Harary, The maximumc onnectivity of a graph, Proc. Nati. Acad. Sci. 4 (1962), 1142–1146.

T.W. Haynes, S.T. Hedetneimi and P.J. Slater, Fundamentals of Domination in Graphs, Marcel-Dekker, Inc., 1997.

KM. Kathiresan and G. Marimuthu, A study on radial graphs, Ars Combin. 96 (2010), 353– 360.

KM. Kathiresan and G. Marimuthu and S.Arockiaraj, Dynamics of radial graphs, Bull. Inst. Combin. Appl. 57 (2009), 21–28.

KM. Kathiresan, S. Arockiaraj, R. Gurusamy and K. Amutha, On the Steiner Radial Number of Graphs, In IWOCA 2012, S. Arumugam and W.F. Smyth (Eds.), Springer-Verlag, Lecture Notes in Comput. Sci. 7643 (2012), 65–72.

O.R. Oellermann and S. Tian, Steiner centers in graphs, J. Graph Theory 14 (5) (1990), 585– 597.

E. Prisner, Graph Dynamics, [Pitmann Research Notes in Mathematics # 338], Longman, London, 1995.

R.R. Singleton, There is no irregular moore graph, Amer. Math. Monthly 7 (1968), 42–43.

### Refbacks

- There are currently no refbacks.

ISSN: 2338-2287

This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.