Characterizing signed graphs which are switching equivalent to their Smarandachely 3-path step signed graphs

Lokesha, V.

Prashanth , B.

Reddy, Siva Kota

English

ZENODO

Scientia MagnaVol. 6 (2010), No. 3, 89-92Smarandachely t-path step signed graphsP. Siva Kota Reddy†, B. Prashanth‡ and V. Lokesha]Department of Mathematics, Acharya Institute of Technology,Bangalore-560 090, IndiaE-mail: pskreddy@acharya.ac.in prashanthb@acharya.ac.in lokeshav@acharya.ac.inAbstract 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. E. Prisner [9] in his book Graph Dynamicsdefines the t-path step operator on the class of finite graphs. Given a graph G and a positiveinteger t, the t-path step graph (G)t of G is formed by taking a copy of the vertex set V (G)of G, joining two vertices u and v in the copy by a single edge e = uv whenever there exists au− v path of length t in G. Analogously, one can define the Smarandachely t-path step signedgraph (S)t = ((G)t, σ′) of a signed graph S = (G, σ) is a signed graph whose underlying graphis (G)t called t-path step graph and sign of any edge e = uv in (S)t is µ(u)µ(v). It is shownthat for any signed graph S, its (S)t is balanced. We then give structural characterizationof Smarandachely t-path step signed graphs. Further, in this paper we characterize signedgraphs which are switching equivalent to their Smarandachely 3-path step signed graphs.Keywords Smarandachely k-signed graphs, Smarandachely k-marked graphs, signed graphs,marked graphs, balance, switching, Smarandachely t-path step signed graphs, negation.§1. IntroductionFor standard terminology and notion in graph theory we refer the reader to Harary [4];the non-standard will be given in this paper as and when required. We treat only finite simplegraphs without self loops and isolates.A Smarandachely k-signed graph (Smarandachely k-marked graph) is an ordered pair S =(G, σ) (S = (G,µ)) where G = (V,E) is a graph called underlying graph of S and σ : E →(e1, e2, · · · , ek) (µ : V → (e1, e2, · · · , ek)) is a function, where each ei ∈ {+,−}. Particularly, aSmarandachely 2-signed graph or Smarandachely 2-marked graph is called abbreviated a signedgraph or a marked graph. A signed graph S = (G, σ) is balanced if every cycle in S has an evennumber of negative edges (Harary [3]). Equivalently a signed graph is balanced if product ofsigns of the edges on every cycle of S is positive.A marking of S is a function µ : V (G) → {+,−}; A signed graph S together with amarking µ by Sµ. Given a signed graph S one can easily define a marking µ of S as follows:90 P. Siva Kota Reddy, B. Prashanth and V. Lokesha No. 3For any vertex v ∈ V (S),µ(v) =∏u∈N(v)σ(uv),the marking µ of S is called canonical marking of S.The following characterization of balanced signed graphs is well known.Proposition 1.1.[6] A signed graph S = (G, σ) is balanced if, and only if, there exist amarking µ of its vertices such that each edge uv in S satisfies σ(uv) = µ(u)µ(v).Given a marking µ of S, by switching S with respect to µ we mean reversing the sign ofevery edge of S whenever the end vertices have opposite signs in Sµ[1] . We denote the signedgraph obtained in this way is denoted by Sµ(S) and this signed graph is called the µ-switchedsigned graph or just switched signed graph. A signed graph S1 switches to a signed graph S2(that is, they are switching equivalent to each other), written S1 ∼ S2, whenever there exists amarking µ such that Sµ(S1) ∼= S2.Two signed graphs S1 = (G, σ) and S2 = (G′, σ′) are said to be weakly isomorphic(Soza´nsky [7] ) or cycle isomorphic (Zaslavsky [8]) if there exists an isomorphism φ : G → G′such that the sign of every cycle Z in S1 equals to the sign of φ(Z) in S2. The following resultis well known:Proposition 1.2.[8] Two signed graphs S1 and S2 with the same underlying graph areswitching equivalent if, and only if, they are cycle isomorphic.§2. Smarandachely t-path step signed graphsGiven a graph G and a positive integer t, the t-path step graph (G)t of G is formed bytaking a copy of the vertex set V (G) of G, joining two vertices u and v in the copy by a singleedge e = uv whenever there exists a u − v path of length t in G. The notion of t-path stepgraphs was defined in [9], page 168.In this paper, we shall now introduce the concept of Smarandachely t-path step signedgraphs as follows: The Smarandachely t-path step signed graph (S)t = ((G)t, σ′) of a signedgraph S = (G, σ) is a signed graph whose underlying graph is (G)t called t-path step graph andsign of any edge e = uv in (S)t is µ(u)µ(v), where µ is the canonical marking of S. Further,a signed graph S = (G, σ) is called Smarandachely t-path step signed graph, if S ∼= (S′)t, forsome signed graph S′.The following result indicates the limitations of the notion of Smarandachely t-path stepsigned graphs as introduced above, since the entire class of unbalanced signed graphs is forbiddento be Smarandachely t-path step signed graphs.Proposition 2.1. For any signed graph S = (G, σ), its (S)t is balanced.Proof. Since sign of any edge e = uv is (S)t is µ(u)µ(v), where µ is the canonical markingof S, by Proposition 1.1, (S)t is balanced.Remark. For any two signed graphs S and S′ with same underlying graph, their Smaran-dachely t-path step signed graphs are switching equivalent.Corollary 2.2. For any signed graph S = (G, σ), its Smarandachely 2 (3)-path step signedgraph (S)2 ((S)3) is balanced.Vol. 6 Smarandachely t-path step signed graphs 91The following result characterize signed graphs which are Smarandachely t-path step signedgraphs.Proposition 2.3. A signed graph S = (G, σ) is a Smarandachely t-path step signed graphif, and only if, S is balanced signed graph and its underlying graph G is a t-path step graph.Proof. Suppose that S is balanced and G is a t-path step graph. Then there exists agraph H such that (H)t ∼= G. Since S is balanced, by Proposition 1.1, there exists a markingµ of G such that each edge e = uv in S satisfies σ(uv) = µ(u)µ(v). Now consider the signedgraph S′ = (H,σ′), where for any edge e in H, σ′(e) is the marking of the corresponding vertexin G. Then clearly, (S′)t ∼= S. Hence S is a Smarandachely t-path step signed graph.Conversely, suppose that S = (G, σ) is a Smarandachely t-path step signed graph. Thenthere exists a signed graph S′ = (H,σ′) such that (S′)t ∼= S. Hence G is the t-path step graphof H and by Proposition 2.1, S is balanced.§3. Switching invariant Smarandachely 3-path step signedgraphsZelinka [9] prove hat the graphs in Fig. 1 are all unicyclic graphs which are fixed in theoperator (G)3, i.e. graphs G such that G ∼= (G)3. The symbols p, q signify that the number ofvertices and edges in Fig. 1.Proposition 3.1.[9] Let G be a finite unicyclic graph such that G ∼= (G)3. Then either Gis a circuit of length not divisible by 3, or it is some of the graphs depicted in Fig. 1.cc ccc ccccsc cc cc cccccccccccp >= 0, q >= 0p qFig.1.In view of the above result, we have the following result for signed graphs:Proposition 3.2. For any signed graph S = (G, σ), S ∼ (S)3 if, and only if, G is a cycleof length not divisible by 3, or it is some of the graphs depicted in Fig. 1 and S is balanced.Proof. Suppose S ∼ (S)3. This implies, G ∼= (G)3 and hence by Proposition 3.1, we seethat the G must be isomorphic to either Cm, 4 ≤ m 6= 3k, k is a positive integer or the graphsdepicted in Fig. 1. Now, if S is any signed graph on any of these graphs, Corollary 4 impliesthat (S)3 is balanced and hence if S is unbalanced its Smarandachely 3-path step signed graph(S)3 being balanced cannot be switching equivalent to S in accordance with Proposition 1.2.Therefore, S must be balanced.92 P. Siva Kota Reddy, B. Prashanth and V. Lokesha No. 3Conversely, suppose that S is a balanced signed graph on Cm, 4 ≤ m 6= 3k, k is a positiveinteger or the graphs depicted in Fig. 1. Then, since (S)3 is balanced as per Corollary 2.2 andsince G ∼= (G)3 in each of these cases, the result follows from Proposition 1.2.Problem. Characterize the signed graphs for which S ∼= (S)3.The notion of negation η(S) of a given signed graph S defined by Harary [3] as follows: η(S)has the same underlying graph as that of S with the sign of each edge opposite to that givento it in S. However, this definition does not say anything about what to do with nonadjacentpairs of vertices in S while applying the unary operator η(.) of taking the negation of S.For a signed graph S = (G, σ), the (S)t is balanced (Proposition 2.1). We now examine,the condition under which negation of (S)t (i.e., η((S)t)) is balanced.Proposition 3.3. Let S = (G, σ) be a signed graph. If (G)t is bipartite then η((S)t) isbalanced.Proof. Since, by Proposition 2.1, (S)t is balanced, then every cycle in (S)t contains evennumber of negative edges. Also, since (G)t is bipartite, all cycles have even length; thus, thenumber of positive edges on any cycle C in (S)t are also even. This implies that the same thingis true in negation of (S)t. Hence η((S)t) is balanced.Proposition 3.2 provides easy solutions to three other signed graph switching equivalencerelations, which are given in the following results.Corollary 3.4. For any signed graph S = (G, σ), η(S) ∼ (S)3 if, and only if, S isunbalanced signed graph on C2m+1, m ≥ 2 or first two graphs depicted in Fig. 1.Corollary 3.5. For any signed graph S = (G, σ), (η(S))3 ∼ (S)3.References[1] R. P. Abelson and M. J. Rosenberg, Symoblic psychologic, A model of attitudinalcognition, Behav. Sci., 3(1958), 1-13.[2] F. Harary, On the notion of balance of a sigraph, Michigan Math. J., 2 (1953), 143-146.[3] F. Harary, Structural duality, Behav. Sci., 2(1957), No. 4, 255-265.[4] F. Harary, Graph Theory, Addison-Wesley Publishing Co., 1969.[5] E. Prisner, Graph Dynamics, Longman House, Burnt Mill, Harlow, 1998.[6] E. Sampathkumar, Point signed and line signed graphs, Nat. Acad. Sci. Letters,7(1984), No. 3, 91-93.[7] T. Soza´nsky, Enueration of weak isomorphism classes of signed graphs, J. Graph Theory,4(1980), No. 2, 127-144.[8] T. Zaslavsky, Signed Graphs, Discrete Appl. Math., 4(1982), No. 1, 47-74.[9] B. Zelinka, The 3-path-step operator on trees and unicyclic graphs, Mathematica Bo-hemica, 127(2002), No. 1, 33-40.

Enueration of weak isomorphism classes of signed graphs,

Graph Dynamics, Longman House,

Graph Theory,

On the notion of balance of a sigraph,

Point signed and line signed graphs,

Signed Graphs,

Structural duality,

Symoblic psychologic, A model of attitudinal cognition,

The 3-path-step operator on trees and unicyclic graphs,

Smarandachely t-path step signed graphs

A Smarandachely k-signed graph (Smarandachely k-marked graph) is an ordered pair S = (G, σ) (S = (G,µ)) where G = (V,E) is a graph called underlying graph of S and σ: 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 called abbreviated a signed graph or a marked graph. E. Prisner [9] in his book Graph Dynamics defines the t-path step operator on the class of finite graphs. Given a graph G and a positive integer t, the t-path step graph (G)t of G is formed by taking a copy of the vertex set V (G) of G, joining two vertices u and v in the copy by a single edge e = uv whenever there exists a u − v path of length t in G. Analogously, one can define the Smarandachely t-path step signed graph (S)t = ((G)t, σ ′) of a signed graph S = (G, σ) is a signed graph whose underlying graph is (G)t called t-path step graph and sign of any edge e = uv in (S)t is µ(u)µ(v). It is shown that for any signed graph S, its (S)t is balanced. We then give structural characterization of Smarandachely t-path step signed graphs. Further, in this paper we characterize signed graphs which are switching equivalent to their Smarandachely 3-path step signed graphs

Smarandachely t-path step signed graphs

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