SOME NEW RESULTS ON INTEGER ADDITIVE SET-VALUED SIGNED GRAPHS

Abstract

Let X denotes a set of non-negative integers and P X be its power set. An integer additive set-labeling IASL of a graph G is an injective set-valued function f : V G → P X − {∅} such that the induced function f + : E G → P X − {∅} is defined by f + uv = f u + f v ; ∀ uv ∈ E G , where f u + f v is the sumset of f u and f v . An IASL of a signed graph is an IASL of its underlying graph G together with the signature σ defined by σ uv = −1 |f+ uv | ; ∀ uv ∈ E Σ . In this paper, we discuss certain characteristics of the signed graphs which admits certain types of integer additive set-labelings.

Keywords

• Signed graphs,balanced signed graphs,clustering of signed graphs,integer additive set-labeled signed graphs,arithmetic integer additive set-labeled signed graphs.

References

1. B. D. Acharya, (2012). Set-valuations of signed digraphs, J. Combin. Inform. System Sci., 37(2- )(2012), 145–167.
2. J Akiyama, D. Avis, V. Cha´vtal and H. Era, (1981). Balancing signed graphs, Discrete App. Math., (4), 227–233., DOI: 10.1016/0166-218X(81)90001-9.
3. P. K. Ashraf, K. A. Germina and N. K. Sudev, A study on set-valuations of signed graphs, Southeast Asian Bull. Math., to appear. J. A. Gallian, (2016). A dynamic survey of graph labeling, Electron. J. Combin., (#DS-6).
4. K. A. Germina and S. Hameed, (2010). On signed paths, signed cycles and their energies, Appl. Math. Sci., 4(70) 3455–3466.
5. K. A. Germina and N. K. Sudev, (2013). On weakly uniform integer additive set-indexers of graphs
6. Int. Math. Forum, 8(37), 1827–1834. DOI: 10.12988/imf.2013.310188.
7. F. Harary, (2001) Graph theory, Narosa Publ. House., New Delhi.
8. F. Harary, (1953). On the notion of balance of a signed graph, The Michigan Math. J., 2(2), 143–146.

9. M. B. Nathanson, (1996). Additive number theory, inverse problems and geometry of sumsets, Springer, New York.
10. N. K. Sudev and K. A. Germina, (2014). On integer additive set-indexers of graphs, Int. J. Math. Sci. Engg. Appl., 8(2)(2014), 11–22.
11. N. K. Sudev and K. A. Germina, A study on arithmetic integer additive set-indexers of graphs, J. Inform. Math. Sci., to appear. N. K. Sudev and K. A. Germina, (2015) On certain types of arithmetic integer additive set-indexers of graphs, Discrete Math. Algorithms Appl., 7(1),1–15., DOI: 10.1142/S1793830915500251.
12. N. K. Sudev and K. A. Germina, (2015). A study on integer additive set-valuation of signed graphs
13. Carpathian Math. Publ., 7(2), 236–246., DOI:10.15330/cmp.7.2.236-246.
14. D. B. West, (2001). Introduction to graph theory, Pearson Education Inc., Delhi.
15. T. Zaslavsky, (1982). Signed graphs, Discrete Appl. Math., 4(1),47–74., DOI: 10.1016/0166- X(82)90033-6.