^{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.

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

2

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.

3

F. Harary (1969). Graph Theory. Addison-Wesley Publishing Co.. , 89-92

4

F. Harary (1953-54). On the notion of balance of a signed graph. Michigan Math. J.. 2, 143-146

5

F. Harary (1955). On local balance and $N$-balance in signed graphs. Michigan Math. J.. 3, 37-41

6

F. Harary, R. Norman and D. Cartwright (1965). Structural models: An introduction to the theory of directed graphs. Jon Wiley, New York.

7

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

8

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

9

R. Rangarajan, M. S. Subramanya and P. Siva Kota Reddy (2012). Neighborhood signed graphs. Southeast
Asian Bulletin of Mathematics. 36 (3), 389-397

10

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

11

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

12

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

13

E. Sampathkumar, P. Siva Kota Reddy, and M. S. Subramanya (2009). Directionally $n$-signed
graphs-II. Int. J. Math. Combin.. 4, 89-98

14

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

15

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

16

P. Siva Kota Reddy, S. Vijay and H. C. Savithri (2010). A note on path sidigraphs. International J.
Math. Combin.. 1, 42-46

17

P. Siva Kota Reddy, S. Vijay and V. Lokesha (2010). n^{th} Power signed graphs-II. International J.
Math. Combin.. 1, 74-79

18

P. Siva Kota Reddy and S. Vijay (2010). Total minimal dominating signed
graph. International J. Math. Combin.. 3, 11-16

19

P. Siva Kota Reddy and K. V. Madhusudhan (2010). Negation switching equivalence in signed graphs. International J. Math. Combin.. 3, 85-90

20

P. Siva Kota Reddy (2010). t-Path sigraphs. Tamsui Oxford
J. of Math. Sciences. 26 (4), 433-441

21

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

22

P. Siva Kota Reddy, B. Prashanth and Kavita. S. Permi (2011). A note on antipodal signed graphs. International J. Math. Combin.. 1, 107-112

23

P. Siva Kota Reddy and B. Prashanth (2012). The common minimal dominating signed graph. Trans. Comb.. 1 (3), 39-46

24

P. Siva Kota Reddy and B. Prashanth (2012). mathcal{S}-Antipodal signed graphs. Tamsui Oxf. J. Inf. Math. Sci.. 28 (2), 165-174

25

P. Siva Kota Reddy, B. Prashanth, and T. R. Vasanth Kumar (2011). Antipodal
signed directed graphs. Advn. Stud. Contemp. Math.. 21 (4), 355-360

26

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

27

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

28

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

29

P. Siva Kota Reddy and U. K. Misra (2013). The equitable associate signed graphs. Bull. Int. Math. Virtual Inst.. 3 (1), 15-20

30

P. Siva Kota Reddy and U. K. Misra (2013). Graphoidal signed graphs. Advn. Stud. Contemp. Math.. 23 (3), 451-460

31

T. Zaslavsky (1982). Signed graphs. Discrete Appl. Math.. 4 (1), 47-74

32

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..