# Sign-Compatibility of Some Derived Signed Graphs

### Abstract

A *signed graph* (or *sigraph* in short) is an ordered pair *S* = (*Su*, *σ*), where* Su* is a graph *G* = (*V*, *E*), called the* underlying graph* of *S* and *σ* : *E* → {+1, −1} is a function from the edge set *E* of *Su* into the set {+1, −1}, called the *signature* of *S*. A sigraph *S* is *sign-compatible* if there exists a *marking* *µ* of its vertices such that the end vertices of every negative edge receive ‘−1’ marks in *µ* and no positive edge does so. In this paper, we characterize *S* such that its ×-line sigraphs, semi-total line sigraphs, semi-total point sigraphs and total sigraphs are sign-compatible.

### References

M Behzad and G T Chartrand, “Line coloring of signed graphs,” Elem. Math., vol. 24, pp. 49-52, 1969.

[2] M Behzad, “A characterization of total graphs,” Proc. Amer. Math. Soc., vol. 26, pp. 383-389, 1970.

[3] M Behzad and H Radjavi, “Structure of regular total graphs,” J. Lond. Math. Soc., vol. 44, pp. 433-436, 1969.

[4] M K Gill, “Contribution to some topics in graph theory and its applications,” Ph.D. Thesis, Indian Institute of Technology, Bombay, 1983.

[5] F Harary, “On the notion of balance of a signed graph,” Michigan Math. J., vol. 2, pp. 143-146, 1953.

[6] F Harary, Graph theory, Massachusetts: Addison-Wesley Publ. Comp., 1969.

[7] E Sampathkumar and S B Chikkodimath, “Semitotal graphs of a graph-I,” J. Karnatak Univ. Sci., vol. 18, pp. 274-280, 1973.

[8] D Sinha, “New frontiers in the theory of signed graph,” Ph.D. Thesis, University of Delhi, Faculty of Technology, 2005.

[9] D Sinha and P Garg, “Balance and consistency of total signed graphs,” Ind. J. Math., vol. 53, pp. 71-81, 2011.

[10] D Sinha and P Garg, “Characterization of total signed graph and semi-total signed graphs,” Int. J. Contemp. Math. Sci., vol. 6, pp. 221-228, 2011.

[11] D Sinha and P Garg, “On the regularity of some signed graph structures,” AKCE Int. J. Graphs Comb., vol. 8, pp. 63-74, 2011.

[12] D Sinha and A Dhama, “Sign-compatibility of common-edge sigraphs and 2-path sigraphs,” Preprint.

[13] D B West, Introduction to graph theory, Prentice-Hall of India Pvt. Ltd., 1996.

[14] T Zaslavsky, “A mathematical bibliography of signed and gain graphs and allied areas,” VII Edition, Electron. J. Combin., #DS8, 1998.

[15] T Zaslavsky, “Glossary of signed and gain graphs and allied areas,” II Edition, Electron. J. Combin., #DS9, 1998.

[2] M Behzad, “A characterization of total graphs,” Proc. Amer. Math. Soc., vol. 26, pp. 383-389, 1970.

[3] M Behzad and H Radjavi, “Structure of regular total graphs,” J. Lond. Math. Soc., vol. 44, pp. 433-436, 1969.

[4] M K Gill, “Contribution to some topics in graph theory and its applications,” Ph.D. Thesis, Indian Institute of Technology, Bombay, 1983.

[5] F Harary, “On the notion of balance of a signed graph,” Michigan Math. J., vol. 2, pp. 143-146, 1953.

[6] F Harary, Graph theory, Massachusetts: Addison-Wesley Publ. Comp., 1969.

[7] E Sampathkumar and S B Chikkodimath, “Semitotal graphs of a graph-I,” J. Karnatak Univ. Sci., vol. 18, pp. 274-280, 1973.

[8] D Sinha, “New frontiers in the theory of signed graph,” Ph.D. Thesis, University of Delhi, Faculty of Technology, 2005.

[9] D Sinha and P Garg, “Balance and consistency of total signed graphs,” Ind. J. Math., vol. 53, pp. 71-81, 2011.

[10] D Sinha and P Garg, “Characterization of total signed graph and semi-total signed graphs,” Int. J. Contemp. Math. Sci., vol. 6, pp. 221-228, 2011.

[11] D Sinha and P Garg, “On the regularity of some signed graph structures,” AKCE Int. J. Graphs Comb., vol. 8, pp. 63-74, 2011.

[12] D Sinha and A Dhama, “Sign-compatibility of common-edge sigraphs and 2-path sigraphs,” Preprint.

[13] D B West, Introduction to graph theory, Prentice-Hall of India Pvt. Ltd., 1996.

[14] T Zaslavsky, “A mathematical bibliography of signed and gain graphs and allied areas,” VII Edition, Electron. J. Combin., #DS8, 1998.

[15] T Zaslavsky, “Glossary of signed and gain graphs and allied areas,” II Edition, Electron. J. Combin., #DS9, 1998.

Published

2012-07-02

How to Cite

SINHA, Deepa; DHAMA, Ayushi.
Sign-Compatibility of Some Derived Signed Graphs.

**Mapana - Journal of Sciences**, [S.l.], v. 11, n. 4, p. 1-14, july 2012. ISSN 0975-3303. Available at: <http://journals.christuniversity.in/index.php/mapana/article/view/277>. Date accessed: 27 may 2019. doi: https://doi.org/10.12723/mjs.23.1.
Section

Research Articles

### Keywords

Sign-compatible, ×-line sigraph, semi-total line sigraph, semi- total point sigraph, total sigraph.