P. Siva Kota Reddy and U. K. Misra

Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132

Volume 18, 2012, Number 4, Pages 40—46

**Download full paper: PDF, 150 Kb**

## Details

### Authors and affiliations

P. Siva Kota Reddy

*Department of Mathematics, Acharya Institute of Technology
Soladevanahalli, Bangalore-560 090, India
* Corresponding author*

U. K. Misra

*Department of Mathematics, Berhampur University
Berhampur-760 007, Orissa, India*

### Abstract

In this paper, we define the common minimal equitable dominating signed graph of a given signed graph and offer a structural characterization of common minimal equitable dominating signed graphs. In the sequel, we also obtained switching equivalence characterization: ~ *CMED*(Σ), where are *CMED*(Σ) are complementary signed graph and common minimal equitable dominating signed graph of Σ respectively

### Keywords

- Signed graphs
- Balance
- Switching
- Complement
- Common minimal equitable dominating signed graph
- Negation

### AMS Classification

- 05C22

### References

- Abelson, R. P., M. J. Rosenberg, Symbolic psychologic: A model of attitudinal cognition, Behav. Sci., Vol. 3, 1958, 1–13.
- Berge, C., Theory of Graphs and its Applications, Methuen, London, 1962.
- Cockayne, E. J., S. T. Hedetniemi, Towards a theory of domination in graphs, Networks,Vol. 7, 1977, 247–261.
- De Jaenisch, C. F., Applications de l’Analyse mathematique an Jen des Echecs, 1862.
- Deepak, G., N. D. Soner, A. Alwardi, Vertex minimal and common minimal equitable dominating graphs, Int. J. Contemp. Math. Sciences, Vol. 7, 2012, No. 10, 499–505.
- Easley, D., J. Kleinberg, Networks, Crowds, and Markets: Reasoning About a Highly Connected World, Cambridge University Press, 2010.
- Harary, F., Graph Theory, Addison-Wesley Publishing Co., 1969.
- Harary, F., On the notion of balance of a signed graph, Michigan Math. J., Vol. 2, 1953,143–146.
- Harary, F., Structural duality, Behav. Sci., Vol. 2, 1957, No. 4, 255–265.
- Ore, O., Theory of Graphs. Amer. Math. Soc. Colloq. Publ., Vol. 38, 1962.
- Rangarajan, R., P. Siva Kota Reddy, The edge
*C*_{4}signed graph of a signed graph, Southeast Asian Bulletin of Mathematics, Vol. 34, 2010, No. 6, 1077–1082. - Rangarajan, R., M. S. Subramanya, P. Siva Kota Reddy, Neighborhood signed graphs, Southeast Asian Bulletin of Mathematics, Vol. 36, 2012, No. 3, 389–397.
- Rouse Ball,W.W., Mathematical Recreation and Problems of Past and Present Times, 1892.
- Sampathkumar, E., Point signed and line signed graphs, Nat. Acad. Sci. Letters, Vol. 7, 1984, No. 3, 91–93.
- Sampathkumar, E., P. Siva Kota Reddy, M. S. Subramanya, Directionally
*n*-signed graphs, Ramanujan Math. Soc., Lecture Notes Series (Proc. Int. Conf. ICDM 2008), Vol. 13, 2010, 155–162. - Sampathkumar, E., P. Siva Kota Reddy, M. S. Subramanya, Directionally
*n*-signed graphs-II, International J. Math. Combin., Vol. 4, 2009, 89–98. - Sampathkumar, E., M. S. Subramanya, P. Siva Kota Reddy, Characterization of line sidigraphs, Southeast Asian Bulletin of Mathematics, Vol. 35, 2011, No. 2, 297–304.
- Siva Kota Reddy, P., M. S. Subramanya, Note on path signed graphs, Notes on Number Theory and Discrete Mathematics, Vol. 15, 2009, No. 4, 1–6.
- Siva Kota Reddy, P., S. Vijay, V. Lokesha,
*n*Power signed graphs, Proceedings of the Jangjeon Math. Soc., Vol. 12, 2009, No. 3, 307–313.^{th} - Siva Kota Reddy, P.,
*t*-Path Sigraphs, Tamsui Oxford J. of Math. Sciences, Vol. 26, 2010, No. 4, 433–441. - Siva Kota Reddy, P., E. Sampathkumar, M. S. Subramanya, Common-edge signed graph of a signed graph, J. Indones. Math. Soc., Vol. 16, 2010, No. 2, 105–112.
- Siva Kota Reddy, P., B. Prashanth, T. R. Vasanth Kumar, Antipodal signed directed Graphs, Advn. Stud. Contemp. Math., Vol. 21, 2011, No. 4, 355–360.
- Siva Kota Reddy, P., B. Prashanth, The Common Minimal Dominating Signed Graph, Trans. Comb., Vol. 1, 2012, No. 3, 39–46.
- Siva Kota Reddy, P., B. Prashanth,
*S*-Antipodal signed graphs, Tamsui Oxford J. of Inf. Math. Sciences, Vol. 28, 2012, No. 2, 165–174. - Siva Kota Reddy, P., S. Vijay, The super line signed graph
*L*(_{r}*S*) of a signed Graph, Southeast Asian Bulletin of Mathematics, Vol. 36, 2012, No. 6, 875–882 - Siva Kota Reddy, P.,d B. Prashanth, Note on Minimal Dominating Signed Graphs, Bull. of Pure & Appl. Math., Vol. 7, 2013, No. 1, to appear.
- Sozánsky, T., Enueration of weak isomorphism classes of signed graphs, J. Graph Theory, Vol. 4, 1980, No. 2, 127–144.
- Yaglom, A. M., I. M. Yaglom. Challenging mathematical problems with elementary solutions. Volume 1: Combinatorial Analysis and Probability Theory, 1964.
- Zaslavsky, T., Signed graphs, Discrete Appl. Math., Vol. 4, 1982, No. 1, 47–74.
- Zaslavsky, T., A mathematical bibliography of signed and gain graphs and allied areas, VIII Edition, Electron. J. Combin., Dynamic Surveys in Combinatorics (1998), No. DS8. Eighth ed. (2012).

## Related papers

## Cite this paper

Siva Kota Reddy, P. & Misra, U. (2012). The common minimal equitable dominating signed graphs, Notes on Number Theory and Discrete Mathematics, 18(4), 40-46.