Directionally $n$-signed graphs-III‎: ‎the notion of symmetric‎ ‎balance

Document Type : Research Paper

Authors

1 Dept. of Mathematics, Siddaganga Institute of Technology, B.H.Road,Tumkur-572103, India.

2 Berhampur University

Abstract

‎Let $G=(V‎, ‎E)$ be a graph‎. ‎By \emph{directional labeling (or‎ ‎d-labeling)} of an edge $x=uv$ of $G$ by an ordered $n$-tuple‎ ‎$(a_1,a_2,\dots,a_n)$‎, ‎we mean a labeling of the edge $x$ such that‎ ‎we consider the label on $uv$ as $(a_1,a_2,\dots,a_n)$ in the‎ ‎direction from $u$ to $v$‎, ‎and the label on $x$ as‎ ‎$(a_{n},a_{n-1},\dots,a_1)$ in the direction from $v$ to $u$‎. ‎In‎ ‎this paper‎, ‎we study graphs‎, ‎called \emph{(n‎,d)-sigraphs}‎, ‎in‎ ‎which every edge is $d$-labeled by an $n$-tuple‎ ‎$(a_1,a_2,\dots,a_n)$‎, ‎where $a_k \in \{+,-\}$‎, ‎for $1\leq k \leq‎ ‎n$‎. ‎In this paper‎, ‎we give different notion of balance‎: ‎symmetric‎ ‎balance in a $(n,d)$-sigraph and obtain some characterizations‎.

Keywords

Main Subjects

References

B. D. Acharya and M. Acharya (1986). New algebraic models of a social system. Indian J. of Pure and Appl. Math.. 17 (2), 152-168 J. Edmonds and E. L. Johnson (1970). Matching: a well-solved class of integral linear programs. in: Richard Guy et al., eds., Combinatorial Structures and Their Applications (Proc. Calgary Int. Conf., Calgary, 1969), Gordon and Breach, New York. F. Harary (1969). Graph Theory. Addison-Wesley Publishing Co.. , 89-92 F. Harary (1953-54). On the notion of balance of a signed graph. Michigan Math. J.. 2, 143-146 F. Harary (1955). On local balance and $N$-balance in signed graphs. Michigan Math. J.. 3, 37-41 F. Harary, R. Norman and D. Cartwright (1965). Structural models: An introduction to the theory of directed graphs. Jon Wiley, New York. R. Rangarajan, M. S. Subramanya and P. Siva Kota Reddy (2010). The H-line signed graph of a signed graph. International J. Math. Combin.. 2, 37-43 R. Rangarajan and P. Siva Kota Reddy (2010). The edge C_4 signed graph of a signed graph. Southeast Asian Bulletin of Mathematics. 34 (6), 1077-1082 R. Rangarajan, M. S. Subramanya and P. Siva Kota Reddy (2012). Neighborhood signed graphs. Southeast Asian Bulletin of Mathematics. 36 (3), 389-397 E. Sampathkumar, P. Siva Kota Reddy and M. S. Subramanya (2008). (3,d)-sigraph and its applications. Advn. Stud. Contemp. Math.. 17 (1), 57-67 E. Sampathkumar, P. Siva Kota Reddy and M. S. Subramanya (2010). (4,d)-sigraph and its applications. Advn. Stud. Contemp. Math.. 20 (1), 115-124 E. Sampathkumar, P. Siva Kota Reddy, and M. S. Subramanya (2010). Directionally $n$-signed graphs, in: B.D. Acharya et al., eds.. Advances in Discrete Mathematics and Applications: Mysore, 2008 (Proc. Int. Conf. Discrete Math., ICDM-2008), Ramanujan, Ramanujan, Math. Soc. Lect. Notes Ser., Ramanujan Mathematical Society, Mysore, India. 13, 153-160 E. Sampathkumar, P. Siva Kota Reddy, and M. S. Subramanya (2009). Directionally $n$-signed graphs-II. Int. J. Math. Combin.. 4, 89-98 P. Siva Kota Reddy and M. S. Subramanya (2009). Signed graph equation L^k(S) \sim \overline{S}. International J. Math. Combin.. 4, 84-88 P. Siva Kota Reddy, S. Vijay and V. Lokesha (2009). n^{th} Power signed graphs. Proceedings of the Jangjeon Math. Soc.. 12 (3), 307-313 P. Siva Kota Reddy, S. Vijay and H. C. Savithri (2010). A note on path sidigraphs. International J. Math. Combin.. 1, 42-46 P. Siva Kota Reddy, S. Vijay and V. Lokesha (2010). n^{th} Power signed graphs-II. International J. Math. Combin.. 1, 74-79 P. Siva Kota Reddy and S. Vijay (2010). Total minimal dominating signed graph. International J. Math. Combin.. 3, 11-16 P. Siva Kota Reddy and K. V. Madhusudhan (2010). Negation switching equivalence in signed graphs. International J. Math. Combin.. 3, 85-90 P. Siva Kota Reddy (2010). t-Path sigraphs. Tamsui Oxford J. of Math. Sciences. 26 (4), 433-441 P. Siva Kota Reddy, E. Sampathkumar and M. S. Subramanya (2010). Common-edge signed graph of a signed graph. J. Indones. Math. Soc.. 16 (2), 105-112 P. Siva Kota Reddy, B. Prashanth and Kavita. S. Permi (2011). A note on antipodal signed graphs. International J. Math. Combin.. 1, 107-112 P. Siva Kota Reddy and B. Prashanth (2012). The common minimal dominating signed graph. Trans. Comb.. 1 (3), 39-46 P. Siva Kota Reddy and B. Prashanth (2012). \mathcal{S}-Antipodal signed graphs. Tamsui Oxf. J. Inf. Math. Sci.. 28 (2), 165-174 P. Siva Kota Reddy, B. Prashanth, and T. R. Vasanth Kumar (2011). Antipodal signed directed graphs. Advn. Stud. Contemp. Math.. 21 (4), 355-360 P. Siva Kota Reddy and U. K. Misra (2012). Common minimal equitable dominating signed graphs. Notes on Number Theory and Discrete Mathematics. 18 (4), 40-46 P. Siva Kota Reddy and S. Vijay (2012). The super line signed graph $\mathcal{L}_r(S)$ of a signed graph. Southeast Asian Bulletin of Mathematics. 36 (6), 875-882 P. Siva Kota Reddy, K. R. Rajanna and Kavita S Permi (2013). The common minimal common neighborhood dominating signed graphs. Trans. Comb.. 2 (1), 1-8 P. Siva Kota Reddy and U. K. Misra (2013). The equitable associate signed graphs. Bull. Int. Math. Virtual Inst.. 3 (1), 15-20 P. Siva Kota Reddy and U. K. Misra (2013). Graphoidal signed graphs. Advn. Stud. Contemp. Math.. 23 (3), 451-460 T. Zaslavsky (1982). Signed graphs. Discrete Appl. Math.. 4 (1), 47-74 T. Zaslavsky (2012). A mathematical bibliography of signed and gain graphs and its allied areas. Electron. J. Combin., Dynamic Surveys in Combinatorics (1998), no. DS8. Eighth ed..
Transactions on Combinatorics
Volume 2, Issue 4 - Serial Number 4
December 2013
Pages 53-62

Files

History

  • Receive Date: 18 August 2013
  • Revise Date: 09 October 2013
  • Accept Date: 10 October 2013
  • Published Online: 01 December 2013

Share

How to cite

Statistics