Unless mentioned or defined otherwise, for all terminology and notion in graph theory the reader is refer to [8]. We consider only finite, simple graphs free from self-loops

Reddy, Siva Kota

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International J.Math. Combin. Vol.3 (2010), 85-90Negation Switching Equivalence in Signed GraphsP.Siva Kota Reddy(Department of Mathematics, Acharya Institute of Technology, Bangalore-560 090, India)K.Shivashankara and K. V.Madhusudhan(Department of Mathematics, Yuvaraja’s College, University of Mysore, Mysore 570 005, India)Email: reddy−math@yahoo.com, drksshankara@gmail.com, kvmadhu13@gmail.comAbstract: A Smarandachely k-signed graph (Smarandachely k-marked graph) is an orderedpair S = (G,σ) (S = (G,µ)) where G = (V, E) is a graph called underlying graph of S andσ : E → (e1, e2, ..., ek) (µ : V → (e1, e2, ..., ek)) is a function, where each ei ∈ {+,−}.Particularly, a Smarandachely 2-signed graph or Smarandachely 2-marked graph is calledabbreviated a signed graph or a marked graph. In this paper, we establish a new graphequation L2(G) ∼= Lk(G), where L2(G) & Lk(G) are second iterated line graph and kthiterated line graph respectively. Further, we characterize signed graphs S for which L2(S) ∼Lk(S) and η(S) ∼ Lk(S), where ∼ denotes switching equivalence and L2(S), Lk(S) andη(S) are denotes the second iterated line signed graph, kth iterated line signed graph andnegation of S respectively.Key Words: Smarandachely k-signed graphs, Smarandachely k-marked graphs, signedgraphs, marked graphs, balance, switching, line signed graphs, negation.AMS(2000): 05C22§1. IntroductionUnless mentioned or defined otherwise, for all terminology and notion in graph theory thereader is refer to [8]. We consider only finite, simple graphs free from self-loops.A Smarandachely k-signed graph (Smarandachely k-marked graph) is an ordered pairS = (G, σ) (S = (G,µ)) where G = (V,E) is a graph called underlying graph of S andσ : E → (e1, e2, ..., ek) (µ : V → (e1, e2, ..., ek)) is a function, where each ei ∈ {+,−}. Particu-larly, a Smarandachely 2-signed graph or Smarandachely 2-marked graph is called abbreviateda signed graph or a marked graph. Cartwright and Harary [5] considered graphs in which ver-tices represent persons and the edges represent symmetric dyadic relations amongst personseach of which designated as being positive or negative according to whether the nature of therelationship is positive (friendly, like, etc.) or negative (hostile, dislike, etc.). Such a network1Received June 1, 2010. Accepted September 15, 2010.86 P.Siva Kota Reddy, K.Shivashankara and K. V.MadhusudhanS is called a signed graph (Chartrand [6]; Harary et al. [11]).Signed graphs are much studied in literature because of their extensive use in modeling avariety socio-psychological process (e.g., see Katai and Iwai [14], Roberts [16] and Roberts andXu [17]) and also because of their interesting connections with many classical mathematicalsystems (Zaslavsky [25]).A cycle in a signed graph S is said to be positive if the product of signs of its edges ispositive. A cycle which is not positive is said to be negative. A signed graph is then said to bebalanced if every cycle in it is positive (Harary [9]). Harary and Kabell [12] developed a simplealgorithm to detect balance in signed graphs as also enumerated them.A marking of S is a function µ : V (G) → {+,−}; A signed graph S together with amarking µ is denoted by Sµ. Given a signed graph S one can easily define a marking µ of S asfollows: For any vertex v ∈ V (S),µ(v) =∏uv∈E(S)σ(uv),the marking µ of S is called canonical marking of S.The following characterization of balanced signed graphs is well known.Theorem 1(E. Sampathkumar, [18]) A signed graph S = (G, σ) is balanced if, and only if,there exists a marking µ of its vertices such that each edge uv in S satisfies σ(uv) = µ(u)µ(v).The idea of switching a signed graph was introduced in [1] in connection with structuralanalysis of social behavior and also its deeper mathematical aspects, significance and connec-tions may be found in [25].Switching S with respect to a marking µ is the operation of changing the sign of every edgeof S to its opposite whenever its end vertices are of opposite signs. The signed graph obtainedin this way is denoted by Sµ(S) and is called µ-switched signed graph or just switched signedgraph. Two signed graphs S1 = (G, σ) and S2 = (G′, σ′) are said to be isomorphic, written asS1 ∼= S2 if there exists a graph isomorphism f : G→ G′ (that is a bijection f : V (G)→ V (G′)such that if uv is an edge in G then f(u)f(v) is an edge in G′) such that for any edge e ∈ G,σ(e) = σ′(f(e)). Further a signed graph S1 = (G, σ) switches to a signed graph S2 = (G′, σ′)(or that S1 and S2 are switching equivalent) written S1 ∼ S2, whenever there exists a markingµ of S1 such that Sµ(S1) ∼= S2. Note that S1 ∼ S2 implies that G ∼= G′, since the definitionof switching does not involve change of adjacencies in the underlying graphs of the respectivesigned graphs.Two signed graphs S1 = (G, σ) and S2 = (G′, σ′) are said to be weakly isomorphic (see[23]) or cycle isomorphic (see [23]) if there exists an isomorphism φ : G → G′ such that thesign of every cycle Z in S1 equals to the sign of φ(Z) in S2. The following result is well known(See [24]).Theorem 2(T. Zaslavsky, [24]) Two signed graphs S1 and S2 with the same underlying graphare switching equivalent if, and only if, they are cycle isomorphic.Negation Switching Equivalence in Signed Graphs 87§2. Negation switching equivalence in signed graphsOne of the important operations on signed graphs involves changing signs of their edges. Fromsocio-psychological point of view, if a signed graph represents the structure of a social systemin which vertices represent persons in a social group, edges represent their pair-wise (dyadic)interactions and sign on each edge represents the qualitative nature of interaction between thecorresponding members in the dyad classified as being positive or negative then according tosocial balance theory, the social system is defined to be in a balanced state if every cycle in thesigned graph contains an even number of negative edges [9]; otherwise, the social system is saidto be in an unbalanced state. The term balance used here refers to the real-life situation inwhich the individuals in a social group experience a state of cognitive stability in the sense thatthere is no psychological tension amongst them that demands a change in the pattern of theirongoing in- terpersonal interactions. For instance, as pointed out by Heider [13], any situationin which a person is forced to maintain a positive relation simultaneously with two other personswho are in conflict with each other is an unbalanced state of the triad consisting of the threepersons. Thus, when the social system is found to be in an unbalanced state it is desired to bringit into a balanced state by means of forcing some positive (negative) relationships change intonegative (positive) relationships; such sets of edges in the corresponding signed graph model arecalled balancing sets (see Katai & Iwai [14]). Such a change (which may be regarded as a unaryoperation transforming the given signed graph) is called negation, which has other implicationsin social psychology too (see Acharya & Joshi [2]). Thus, formally, the negation η(S) of S is asigned graph obtained from S by negating the sign of every edge of S; that is, by changing thesign of each edge to its opposite [10].Behzad and Chartrand [4] introduced the notion of line signed graph L(S) of a given signedgraph S as follows: Given a signed graph S = (G, σ) its line signed graph L(S) = (L(G), σ′)is the signed graph whose underlying graph is L(G), the line graph of G, where for any edgeeiej in L(S), σ′(eiej) is negative if, and only if, both ei and ej are adjacent negative edges inS. Another notion of line signed graph introduced in [7] is as follows: The line signed graph ofa signed graph S = (G, σ) is a signed graph L(S) = (L(G), σ′), where for any edge ee′ in L(S),σ′(ee′) = σ(e)σ(e′). In this paper, we follow the notion of line signed graph defined by M. K.Gill [7] (See also E. Sampathkumar et al. [19,20]).Theorem 3(M. Acharya, [3]) For any signed graph S = (G, σ), its line signed graph L(S) =(L(G), σ′) is balanced.Hence, we shall call a given signed graph S a line signed graph if it is isomorphic to theline signed graph L(S′) of some signed graph S′. In [20], the authors obtained a structuralcharacterization of line signed graphs as well as line signed digraphs.For any positive integer k, the kth iterated line graph, Lk(G) of G (kth iterated line signedgraph, Lk(S) of S) is defined as follows:L0(G) = G, Lk(G) = L(Lk−1(G)) (L0(S) = S, Lk(S) = L(Lk−1(S)))Corollary 4(P. Siva Kota Reddy & M. S. Subramanya, [22]) For any signed graph S = (G, σ)88 P.Siva Kota Reddy, K.Shivashankara and K. V.Madhusudhanand for any positive integer k, Lk(S) is balanced.The following result is well known.Theorem 5(V. V. Menon, [15]) For a graph G, G ∼= Lk(G) for any integers k ≥ 1 if, and onlyif, G is 2-regular.Proposition 6(D. Sinha, [21]) For a connected graph G = (V,E), L(G) ∼= L2(G) if, and onlyif, G is cycle or K1,3.From the above results we have the following result for graphs.Theorem 7 For any graph G, L2(G) ∼= Lk(G) for some k ≥ 3, if, and only if, G is either acycle or K1,3.Proof Suppose that L2(G) ∼= Lk(G) for some k ≥ 3. We observe that Lk(G) = Lk−2(L2(G)).Hence, by Proposition 6, L2(G) must be a cycle. But for any graph G, L(G) is a cycle if, andonly if, G is either cycle or K1,3. Since K1,3 is a forbidden to line graph and L(G) is a linegraph, G 6= K1,3. Hence L(G) must be a cycle. Finally L(G) is a cycle if, and only if, G iseither a cycle or K1,3.Conversely, if G is a cycle Cr , of length r, r ≥ 3 then for any k ≥ 2, Lk(G) is a cycle andif G = K1,3 then for any k ≥ 2, Lk(G) = C3. This implies, L2(G) = Lk(G) for any k ≥ 3. Thiscompletes the proof. We now characterize those second iterated line signed graphs that are switching equivalentto their kth iterated line signed graphs.Proposition 8 For any signed graph S = (G, σ), L2(S) ∼ Lk(S) if, and only if, G is either acycle or K1,3.Proof Suppose L2(S) ∼ Lk(S). This implies, L2(G) ∼= Lk(G) and hence by Theorem 7,we see that the graph G must be isomorphic to either a cycle or K1,3.Conversely, suppose that G is a cycle or K1,3. Then L2(G) ∼= Lk(G) by Theorem 7. Now,if S any signed graph on any of these graphs, By Corollary 4, L2(S) and Lk(S) are balancedand hence, the result follows from Theorem 2. We now characterize those negation signed graphs that are switching equivalent to theirline signed graphs.Proposition 9 For any signed graph S = (G, σ), η(S) ∼ Lk(S) if, and only if, S is anunbalanced signed graph and G is 2-regular with odd length.Proof Suppose η(S) ∼ Lk(S). This implies, G ∼= Lk(G) and hence G is 2-regular. Now,if S is any signed graph with underlying graph as 2-regular, Corollary 4 implies that Lk(S) isbalanced. Now if S is an unbalanced signed graph with underlying graph G = Cn, where n iseven, then clearly η(S) is unbalanced. Next, if S is unbalanced signed graph with underlyinggraph G = Cn, where n is odd, then η(S) is unbalanced. Hence, if η(S) is unbalanced andNegation Switching Equivalence in Signed Graphs 89its line signed graph Lk(S) being balanced can not be switching equivalent to S in accordancewith Theorem 2. Therefore, S must be unbalanced and G is 2-regular with odd length.Conversely, suppose that S is an unbalanced signed graph and G is 2-regular with oddlength. Then, since Lk(S) is balanced as per Corollary 4 and since G ∼= Lk(G), the resultfollows from Theorem 2 again. Corollary 10 For any signed graph S = (G, σ), η(S) ∼ L(S) if, and only if, S is an unbalancedsigned graph and G is 2-regular with odd length.AcknowledgementThe first author very much thankful to Sri. B. Premnath Reddy, Chairman, Acharya Institutes,for his constant support and encouragement for R & D.References[1] R. P. Abelson and M. J. Rosenberg, Symoblic psychologic:A model of attitudinal cognition,Behav. Sci., 3 (1958), 1-13.[2] B. D. Acharya and S. Joshi, Semibalance in signed digraphs, Proc. of the InternationalConference on Recent Trends and New Directions of Research in Cybernetics and Systems(January, 1-3, 2004), Institute of Advanced Study in Science & Technology, Guwahati(India).[3] M. Acharya, x-Line sigraph of a sigraph, J. Combin. Math. Combin. Comput., 69(2009),103-111.[4] M. Behzad and G. T. Chartrand, Line-coloring of signed graphs, Elemente der Mathematik,24(3) (1969), 49-52.[5] D.W. Cartwright and F. Harary, Structural balance: A generalization of Heider’s Theory,Psych. Rev., 63(1956), 277-293.[6] G.T. Chartrand, Graphs as Mathematical Models, Prindle, Weber & Schmidt, Inc., Boston,Massachusetts 1977.[7] M. K. Gill, Contributions to some topics in graph theory and its applications, Ph.D. thesis,The Indian Institute of Technology, Bombay, 1983.[8] F. Harary, Graph Theory, Addison-Wesley Publishing Co., 1969.[9] F. Harary, On the notion of balance of a signed graph,Michigan Math. J., 2(1953), 143-146.[10] F. Harary, Structural duality, Behav. Sci., 2(4) (1957), 255-265.[11] F. Harary, R.Z. Norman and D.W. Cartwright, Structural models: An introduction to thetheory of directed graphs, Wiley Inter-Science, Inc., New York, 1965.[12] F. Harary and J.A. Kabell, Counting balanced signed graphs using marked graphs, Proc.Edinburgh Math. Soc., 24 (2)(1981), 99-104.[13] F. Heider, Attitudes and cognitive organization, J. Psychol., 21(1946), 107-112.[14] O. Katai and S. Iwai, Studies on the balancing, the minimal balancing, and the minimumbalancing processes for social groups with planar and nonplanar graph structures, J. Math.Psychol., 18(1978), 140-176.90 P.Siva Kota Reddy, K.Shivashankara and K. V.Madhusudhan[15] V. V. Menon, On repeated interchange graphs, Amer. Math. Monthly, 13(1996), 986-989.[16] F.S. Roberts, Graph Theory and its Applications to Problems of Society, SIAM, Philadel-phia, PA, USA, 1978.[17] F.S. Roberts and Shaoji Xu, Characterizations of consistent marked graphs, Discrete Ap-plied Mathematics, 127(2003), 357- 371.[18] E. Sampathkumar, Point signed and line signed graphs, Nat. Acad. Sci. Letters, 7(3)(1984), 91-93.[19] E. Sampathkumar, P. Siva Kota Reddy, and M. S. Subramanya, The Line n-sigraph of asymmetric n-sigraph, Southeast Asian Bulletin of Mathematics, 34(4) (2010), to appear.[20] E. Sampathkumar, M. S. Subramanya and P. Siva Kota Reddy, Characterization of LineSidigraphs, Southeast Asian Bull. Math.(Springer-Verlag), to appear.[21] D. Sinha, New frontiers in the theory of signed graphs, Ph.D. Thesis, University of Delhi,2005.[22] P. Siva Kota Reddy, and M. S. Subramanya, Signed Graph Equation LK(S) ∼ S, Inter-national J. Math. Combin., 4 (2009), 84-88.[23] T. Soza´nsky, Enueration of weak isomorphism classes of signed graphs, J. Graph Theory,4(2)(1980), 127-144.[24] T. Zaslavsky, Signed Graphs, Discrete Appl. Math., 4(1)(1982), 47-74.[25] T. Zaslavsky, A mathematical bibliography of signed and gain graphs and its allied areas,Electronic J. Combin., 8(1)(1998), Dynamic Surveys (1999), No. DS8.

85-90 Negation Switching Equivalence in Signed Graphs P.Siva Kota Reddy (Department of Mathematics, Acharya Institute of Technology, Bangalore-560 090, India) K.Shivashankara and K. V.Madhusudhan

A mathematical bibliography of signed and gain graphs and its allied areas,

Attitudes and cognitive organization,

Characterizations of consistent marked graphs,

Contributions to some topics in graph theory and its applications,

Counting balanced signed graphs using marked graphs,

Enueration of weak isomorphism classes of signed graphs,

Graph Theory and its Applications to Problems of Society,

Graph Theory,

Graphs as Mathematical Models,

Line-coloring of signed graphs,

New frontiers in the theory of signed graphs,

On repeated interchange graphs,

On the notion of balance of a signed graph,Michigan

Point signed and line signed graphs,

Semibalance in signed digraphs,

Signed Graphs,

Structural balance: A generalization of Heider’s Theory,

Structural duality,

Structural models: An introduction to the theory of directed graphs,

Studies on the balancing, the minimal balancing, and the minimum balancing processes for social groups with planar and nonplanar graph structures,

Negation Switching Equivalence in Signed Graphs

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