Introducing a new notion S-antipodal symmetric n-sigraph of a symmetric n-sigraph and its properties are obtained. Also giving the relation between antipodal symmetric n-sigraphs and S-antipodal symmetric n-sigraphs. Further, discussing structural characterization of S-antipodal symmetric n-sigraphs

Geetha, M.C.

Rajanna, K.R.

Reddy, Siva Kota

English

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Switching Equivalence in Symmetric n-Sigraphs-V

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International J.Math. Combin. Vol.3(2012), 58-63Switching Equivalence in Symmetric n-Sigraphs-VP.Siva Kota Reddy, M.C.Geetha and K.R.Rajanna(Department of Mathematics, Acharya Institute of Technology, Soldevanahalli, Bangalore-560 090, India)E-mail: pskreddy@acharya.ac.inAbstract: An n-tuple (a1, a2, ..., an) is symmetric, if ak = an−k+1, 1 ≤ k ≤ n. LetHn = {(a1, a2, · · · , an) : ak ∈ {+,−}, ak = an−k+1, 1 ≤ k ≤ n} be the set of all symmetricn-tuples. A symmetric n-sigraph (symmetric n-marked graph) is an ordered pair Sn = (G,σ)(Sn = (G,µ)), where G = (V,E) is a graph called the underlying graph of Sn and σ : E → Hn(µ : V → Hn) is a function. In this paper, we introduced a new notion S-antipodal symmetricn-sigraph of a symmetric n-sigraph and its properties are obtained. Also we give the relationbetween antipodal symmetric n-sigraphs and S-antipodal symmetric n-sigraphs. Further, wediscuss structural characterization of S-antipodal symmetric n-sigraphs.Key Words: Symmetric n-sigraphs, Smarandachely symmetric n-marked graph, sym-metric n-marked graphs, balance, switching, antipodal symmetric n-sigraphs, S-antipodalsymmetric n-sigraphs, complementation.AMS(2010): 05C22§1. IntroductionUnless mentioned or defined otherwise, for all terminology and notion in graph theory thereader is refer to [1]. We consider only finite, simple graphs free from self-loops.Let n ≥ 1 be an integer. An n-tuple (a1, a2, · · · , an) is symmetric, if ak = an−k+1, 1 ≤k ≤ n. Let Hn = {(a1, a2, · · · , an) : ak ∈ {+,−}, ak = an−k+1, 1 ≤ k ≤ n} be the set of allsymmetric n-tuples. Note that Hn is a group under coordinate wise multiplication, and theorder of Hn is 2m, where m = ⌈n2⌉.A Smarandachely k-marked graph (Smarandachely k-signed graph) is an ordered pairS = (G,µ) (S = (G, σ)) where G = (V,E) is a graph called underlying graph of S andµ : V → {e1, e2, ..., ek} (σ : E → {e1, e2, ..., ek}) is a function, where ei ∈ {+,−}. An n-tuple (a1, a2, ..., an) is symmetric, if ak = an−k+1, 1 ≤ k ≤ n. Let Hn = {(a1, a2, ..., an) :ak ∈ {+,−}, ak = an−k+1, 1 ≤ k ≤ n} be the set of all symmetric n-tuples. A Smaran-dachely symmetric n-marked graph (Smarandachely symmetric n-signed graph) is an orderedpair Sn = (G,µ) (Sn = (G, σ)) where G = (V,E) is a graph called the underlying graph of Snand µ : V → Hn (σ : E → Hn) is a function. Particularly, a Smarandachely 1-marked graph(Smarandachely 1-signed graph) is called a marked graph (signed graph).1Received February 25, 2012. Accepted September 9, 2012.Switching Equivalence in Symmetric n-Sigraphs-V 59In this paper by an n-tuple/n-sigraph/n-marked graph we always mean a symmetric n-tuple/symmetric n-sigraph/symmetric n-marked graph.An n-tuple (a1, a2, · · · , an) is the identity n-tuple, if ak = +, for 1 ≤ k ≤ n, otherwiseit is a non-identity n-tuple. In an n-sigraph Sn = (G, σ) an edge labelled with the identityn-tuple is called an identity edge, otherwise it is a non-identity edge. Further, in an n-sigraphSn = (G, σ), for any A ⊆ E(G) the n-tuple σ(A) is the product of the n-tuples on the edges ofA.In [7], the authors defined two notions of balance in n-sigraph Sn = (G, σ) as follows (Seealso R. Rangarajan and P.S.K.Reddy [4]):Definition 1.1 Let Sn = (G, σ) be an n-sigraph. Then,(i) Sn is identity balanced (or i-balanced), if product of n-tuples on each cycle of Sn isthe identity n-tuple, and(ii) Sn is balanced, if every cycle in Sn contains an even number of non-identity edges.Note 1.1 An i-balanced n-sigraph need not be balanced and conversely.The following characterization of i-balanced n-sigraphs is obtained in [7].Proposition 1.1 (E. Sampathkumar et al. [7]) An n-sigraph Sn = (G, σ) is i-balanced if, andonly if, it is possible to assign n-tuples to its vertices such that the n-tuple of each edge uv isequal to the product of the n-tuples of u and v.Let Sn = (G, σ) be an n-sigraph. Consider the n-marking µ on vertices of Sn defined asfollows: each vertex v ∈ V , µ(v) is the n-tuple which is the product of the n-tuples on theedges incident with v. Complement of Sn is an n-sigraph Sn = (G, σc), where for any edgee = uv ∈ G, σc(uv) = µ(u)µ(v). Clearly, Sn as defined here is an i-balanced n-sigraph due toProposition 1.1 ([10]).In [7], the authors also have defined switching and cycle isomorphism of an n-sigraphSn = (G, σ) as follows (See also [2,5,6,10]):Let Sn = (G, σ) and S′n = (G′, σ′), be two n-sigraphs. Then Sn and S′n are said to beisomorphic, if there exists an isomorphism φ : G → G′ such that if uv is an edge in Sn withlabel (a1, a2, · · · , an) then φ(u)φ(v) is an edge in S′n with label (a1, a2, · · · , an).Given an n-marking µ of an n-sigraph Sn = (G, σ), switching Sn with respect to µ isthe operation of changing the n-tuple of every edge uv of Sn by µ(u)σ(uv)µ(v). The n-sigraphobtained in this way is denoted by Sµ(Sn) and is called the µ-switched n-sigraph or just switchedn-sigraph. Further, an n-sigraph Sn switches to n-sigraph S′n (or that they are switchingequivalent to each other), written as Sn ∼ S′n, whenever there exists an n-marking of Sn suchthat Sµ(Sn) ∼= S′n.Two n-sigraphs Sn = (G, σ) and S′n = (G′, σ′) are said to be cycle isomorphic, if thereexists an isomorphism φ : G→ G′ such that the n-tuple σ(C) of every cycle C in Sn equals tothe n-tuple σ(φ(C)) in S′n. We make use of the following known result (see [7]).Proposition 1.2 (E. Sampathkumar et al. [7]) Given a graph G, any two n-sigraphs with G60 P.Siva Kota Reddy, M.C.Geetha and K.R.Rajannaas underlying graph are switching equivalent if, and only if, they are cycle isomorphic.Let Sn = (G, σ) be an n-sigraph. Consider the n-marking µ on vertices of S definedas follows: each vertex v ∈ V , µ(v) is the product of the n-tuples on the edges incidentat v. Complement of S is an n-sigraph Sn = (G, σ′), where for any edge e = uv ∈ G,σ′(uv) = µ(u)µ(v). Clearly, Sn as defined here is an i-balanced n-sigraph due to Proposition1.1.§2. S-Antipodal n-SigraphsRadhakrishnan Nair and Vijayakumar [3] has introduced the concept of S-antipodal graph of agraph G as the graph A∗(G) has the vertices in G with maximum eccentricity and two verticesof A∗(G) are adjacent if they are at a distance of diam(G) in G.Motivated by the existing definition of complement of an n-sigraph, we extend the notionof S-antipodal graphs to n-sigraphs as follows:The S-antipodal n-sigraph A∗(Sn) of an n-sigraph Sn = (G, σ) is an n-sigraph whoseunderlying graph is A∗(G) and the n-tuple of any edge uv is A∗(Sn) is µ(u)µ(v), where µ is thecanonical n-marking of Sn. Further, an n-sigraph Sn = (G, σ) is called S-antipodal n-sigraph, ifSn ∼= A∗(S′n) for some n-sigraph S′n. The following result indicates the limitations of the notionA∗(Sn) as introduced above, since the entire class of i-unbalanced n-sigraphs is forbidden to beS-antipodal n-sigraphs.Proposition 2.1 For any n-sigraph Sn = (G, σ), its S-antipodal n-sigraph A∗(Sn) is i-balanced.Proof Since the n-tuple of any edge uv in A∗(Sn) is µ(u)µ(v), where µ is the canonicaln-marking of Sn, by Proposition 1.1, A∗(Sn) is i-balanced. For any positive integer k, the kth iterated S-antipodal n-sigraph A∗(Sn) of Sn is definedas follows:(A∗)0(Sn) = Sn, (A∗)k(Sn) = A∗((A∗)k−1(Sn))Corollary 2.2 For any n-sigraph Sn = (G, σ) and any positive integer k, (A∗)k(Sn) is i-balanced.In [3], the authors characterized those graphs that are isomorphic to their S-antipodalgraphs.Proposition 2.3(Radhakrishnan Nair and Vijayakumar [3]) For a graph G = (V,E), G ∼=A∗(G) if, and only if, G is a regular self-complementary graph.We now characterize the n-sigraphs that are switching equivalent to their S-antipodaln-sigraphs.Proposition 2.4 For any n-sigraph Sn = (G, σ), Sn ∼ A∗(Sn) if, and only if, G is regularSwitching Equivalence in Symmetric n-Sigraphs-V 61self-complementary graph and Sn is i-balanced n-sigraph.Proof Suppose Sn ∼ A∗(Sn). This implies, G ∼= A∗(G) and hence G is is a regularself-complementary graph. Now, if Sn is any n-sigraph with underlying graph as regular self-complementary graph, Proposition 2.1 implies that A∗(Sn) is i-balanced and hence if S is i-unbalanced and its A∗(Sn) being i-balanced can not be switching equivalent to Sn in accordancewith Proposition 1.2. Therefore, Sn must be i-balanced.Conversely, suppose that Sn is an i-balanced n-sigraph andG is regular self-complementary.Then, since A∗(Sn) is i-balanced as per Proposition 2.1 and since G ∼= A∗(G), the result followsfrom Proposition 1.2 again. Proposition 2.5 For any two vs Sn and S′n with the same underlying graph, their S-antipodaln-sigraphs are switching equivalent.Remark 2.6 If G is regular self-complementary graph, then G ∼= G. The above result is holdsgood for Sn ∼ A∗(Sn).In [16], P.S.K.Reddy et al. introduced antipodal n-sigraph of an n-sigraph as follows:The antipodal n-sigraph A(Sn) of an n-sigraph Sn = (G, σ) is an n-sigraph whose un-derlying graph is A(G) and the n-tuple of any edge uv in A(Sn) is µ(u)µ(v), where µ is thecanonical n-marking of Sn. Further, an n-sigraph Sn = (G, σ) is called antipodal n-sigraph, ifSn ∼= A(S′n) for some n-sigraph S′n.Proposition 2.7(P.S.K.Reddy et al. [16]) For any n-sigraph Sn = (G, σ), its antipodal n-sigraph A(Sn) is i-balanced.We now characterize n-sigraphs whose S-antipodal n-sigraphs and antipodal n-sigraphsare switching equivalent. In case of graphs the following result is due to Radhakrishnan Nairand Vijayakumar [3].Proposition 2.8 For a graph G = (V,E), A∗(G) ∼= A(G) if, and only if, G is self-centred.Proposition 2.9 For any n-sigraph Sn = (G, σ), A∗(Sn) ∼ A(Sn) if, and only if, G is self-centred.Proof Suppose A∗(Sn) ∼ A(Sn). This implies, A∗(G) ∼= A(G) and hence by Proposition2.8, we see that the graph G must be self-centred.Conversely, suppose that G is self centred. Then A∗(G) ∼= A(G) by Proposition 2.8. Now,if Sn is an n-sigraph with underlying graph as self centred, by Propositions 2.1 and 2.7, A∗(Sn)and A(Sn) are i-balanced and hence, the result follows from Proposition 1.2.In [3], the authors shown that A∗(G) ∼= A∗(G) if G is either complete or totally discon-nected. We now characterize n-sigraphs whose A∗(Sn) and A∗(Sn) are switching equivalent.Proposition 2.10 For any signed graph S = (G, σ), A∗(Sn) ∼ A∗(Sn) if, and only if, G iseither complete or totally disconnected.62 P.Siva Kota Reddy, M.C.Geetha and K.R.RajannaThe following result characterize n-sigraphs which are S-antipodal n-sigraphs.Proposition 2.11 An n-sigraph Sn = (G, σ) is a S-antipodal n-sigraph if, and only if, Sn isi-balanced n-sigraph and its underlying graph G is a S-antipodal graph.Proof Suppose that Sn is i-balanced and G is a A(G). Then there exists a graph Hsuch that A∗(H) ∼= G. Since Sn is i-balanced, by Proposition 1.1, there exists an n-markingµ of G such that each edge uv in Sn satisfies σ(uv) = µ(u)µ(v). Now consider the n-sigraphS′n = (H,σ′), where for any edge e in H , σ′(e) is the n-marking of the corresponding vertex inG. Then clearly, A∗(S′n)∼= Sn. Hence Sn is a S-antipodal n-sigraph.Conversely, suppose that Sn = (G, σ) is a S-antipodal n-sigraph. Then there exists ann-sigraph S′n = (H,σ′) such that A∗(S′n)∼= Sn. Hence G is the A∗(G) of H and by Proposition2.1, Sn is i-balanced. §3. ComplementationIn this section, we investigate the notion of complementation of a graph whose edges have signs(a sigraph) in the more general context of graphs with multiple signs on their edges. We lookat two kinds of complementation: complementing some or all of the signs, and reversing theorder of the signs on each edge.For any m ∈ Hn, the m-complement of a = (a1, a2, · · · ., an) is: am = am. For anyM ⊆ Hn, and m ∈ Hn, the m-complement of M is Mm = {am : a ∈ M}. For any m ∈ Hn,the m-complement of an n-sigraph Sn = (G, σ), written (Smn ), is the same graph but with eachedge label a = (a1, a2, ..., an) replaced by am. For an n-sigraph Sn = (G, σ), the A∗(Sn) isi-balanced (Proposition 2.1). We now examine, the condition under which m-complement ofA(Sn) is i-balanced, where for any m ∈ Hn.Proposition 3.1 Let Sn = (G, σ) be an n-sigraph. Then, for any m ∈ Hn, if A∗(G) isbipartite then (A∗(Sn))m is i-balanced.Proof Since, by Proposition 2.1, A∗(Sn) is i-balanced, for each k, 1 ≤ k ≤ n, the numberof n-tuples on any cycle C in A∗(Sn) whose kth co-ordinate are − is even. Also, since A∗(G)is bipartite, all cycles have even length; thus, for each k, 1 ≤ k ≤ n, the number of n-tuples onany cycle C in A∗(Sn) whose kth co-ordinate are + is also even. This implies that the samething is true in any m-complement, where for any m,∈ Hn. Hence (A∗(Sn))t is i-balanced. Problem 3.2 Characterize these n-sigraphs for which(1) (Sn)m ∼ A∗(Sn);(2) (Sn)m ∼ A(Sn);(3) (A∗(Sn))m ∼ A(Sn);(4) A∗(Sn) ∼ (A(Sn))m;(5) (A∗(S))m ∼ A∗(Sn);(6) A∗(Sn) ∼ (A∗(Sn))m.Switching Equivalence in Symmetric n-Sigraphs-V 63References[1] F. Harary, Graph Theory, Addison-Wesley Publishing Co., 1969.[2] V.Lokesha, P.S.K.Reddy and S. Vijay, The triangular line n-sigraph of a symmetric n-sigraph, Advn. Stud. Contemp. 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Switching equivalence in symmetric n-sigraphs-III,

Switching Equivalence in Symmetric n-Sigraphs-V P.Siva Kota Reddy, M.C.Geetha and K.R.Rajanna

Switching Equivalence in Symmetric n-Sigraphs-V

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