For standard terminology and notion in graph theory we refer the reader to Harary; the non-standard will be given in this paper as and when required. We treat only finite simple graphs without self loops and isolates

Rangarajan, R.

Reddy, Siva Kota

Subramanya, M.S.

English

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International J.Math. Combin. Vol.2 (2010), 37-43The H-Line Signed Graph of a Signed GraphR.Rangarajan and M. S. SubramanyaDepartment of Studies in Mathematics of University of MysoreManasagangotri, Mysore 570 006, IndiaP. Siva Kota ReddyDepartment of Mathematics of Acharya Institute of Graduate StudiesSoldevanahalli, Bangalore 560 090, IndiaEmail: reddy−math@yahoo.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. Given a connected graph H of order atleast 3, the H-Line Graph of a graph G = (V,E), denoted by HL(G), is a graph with thevertex set E, the edge set of G where two vertices in HL(G) are adjacent if, and only if, thecorresponding edges are adjacent in G and there exists a copy of H in G containing them.Analogously, for a connected graph H of order at lest 3, we define the H-Line signed graphHL(S) of a signed graph S = (G,σ) as a signed graph, HL(S) = (HL(G), σ′), and for anyedge e1e2 in HL(S), σ′(e1e2) = σ(e1)σ(e2). In this paper, we characterize signed graphs Swhich are H-line signed graphs and study some properties of H-line graphs as well as H-linesigned graphs.Key Words: Smarandachely k-Signed graphs, Smarandachely k-Marked graphs, Signedgraphs, Balance, Switching, H-Line signed graph.AMS(2000): 05C22§1. IntroductionFor standard terminology and notion in graph theory we refer the reader to Harary [8]; thenon-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 signed1Received May 4, 2010. Accepted June 8, 2010.38 R.Rangarajan, M. S. Subramanya and P. Siva Kota Reddygraph or a marked graph. We say that a signed graph is connected if its underlying graph isconnected. A signed graph S = (G, σ) is balanced if every cycle in S has an even number ofnegative edges (See [9]). Equivalently a signed graph is balanced if product of signs of the edgeson every cycle of S is positive.A marking of S is a function µ : V (G) → {+,−}; A signed graph S together with amarking µ is denoted by Sµ.The following characterization of balanced signed graphs is well known.Theorem 1.1(E. Sampathkumar [12]) 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).Given a signed graph S one can easily define a marking µ of S as follows: For any vertexv ∈ V (S),µ(v) =∏uv∈E(S)σ(uv),the marking µ of S is called canonical marking of S.The idea of switching a signed graph was introduced by Abelson and Rosenberg [1] inconnection with structural analysis of marking µ of a signed graph S. Switching S with respectto a marking µ is the operation of changing the sign of every edge of S to its opposite wheneverits end vertices are of opposite signs. The signed graph obtained in this way is denoted bySµ(S) and is called µ-switched signed graph or just switched signed graph. Two signed graphsS1 = (G, σ) and S2 = (G′, σ′) are said to be isomorphic, written as S1 ∼= S2 if there existsa graph isomorphism f : G → G′ (that is a bijection f : V (G) → V (G′) such that if uv isan 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 S2are switching equivalent) written S1 ∼ S2, whenever there exists a marking µ of S1 such thatSµ(S1) ∼= S2. Note that S1 ∼ S2 implies that G ∼= G′, since the definition of switching doesnot involve change of adjacencies in the underlying graphs of the respective signed graphs.Two signed graphs S1 = (G, σ) and S2 = (G′, σ′) are said to be weakly isomorphic (see[22]) 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 [23]):Theorem 1.2(T. Zaslavsky [23]) Two signed graphs S1 and S2 with the same underlying graphare switching equivalent if, and only if, they are cycle isomorphic.§2. H-Line Signed Graph of a Signed GraphThe line graph L(G) of a nonempty graph G = (V,E) is the graph whose vertices are the edgesof G and two vertices are adjacent if and only if the corresponding edges are adjacent. Thetriangular line graph T (G) of a nonempty graph was introduced by Jerret [10] as a graph whosevertices are edges of G and two vertices are adjacent if and only if corresponding edges belongsto a common triangle. Triangular graphs were introduced to model a metric space defined onThe H-Line Signed Graph of a Signed Graph 39the edge set of a graph. These concepts were generalized in [5] as follows: Let H be a fixedconnected graph of order at least 3. For a graph G of size the H-line graph of G, denotedby HL(G),is the graph whose vertices are the edges of G and two vertices are adjacent thecorresponding edges are adjacent and belong to a copy of H .If H ∼= P3 then HL(G) = L(G)and so H-line graph is a generalization of line graphs. Clearly, if a graph is H free, then itsH-line graph is trivial.In [10], the authors introduced the notion of triangular line graph of a graph as follows:The triangular line graph of a G = (V,E) denoted by T (G) = (V ′, E′), whose vertices are theedges of G and two vertices are adjacent the corresponding edges belongs to a triangle in G.Clearly for any graph G, T (G) = K3L(G).Behzad and Chartrand [3] introduced the notion of line signed graph L(S) of a given signedgraph S as follows: L(S) is a signed graph such that (L(S))u ∼= L(Su) and an edge eiej in L(S)is negative if, and only if, both ei and ej are adjacent negative edges in S. Another notion of linesigned graph introduced in [7], is as follows: The line signed graph of a 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′). Inthis paper, we follow the notion of line signed graph defined by M. K. Gill [7] (See also E.Sampathkumar et al. [13,14]). For more operations on signed graphs see [15-20].Proposition 2.1 For any signed graph S = (G, σ), its line signed graph L(S) = (L(G), σ′) isbalanced.In [21], the authors extends the notion of triangular line graphs to triangular line signedgraphs. We now extend the notion of H-line graph to the realm of signed graph as follows:Let S = (G, σ) ba a signed graph. For any fixed connected graph H of order at least 3,the H-line signed graph of S, denoted by HL(S) is the signed graph HL(S) = (HL(G), σ′)whose underlying graph is HL(G) and for any edge ee′ in HL(G), σ′(ee′) = σ(e)σ(e′). Furthera signed graph S is said to be H-line signed graph if there exists a signed graph S′ such thatHL(S′) ∼= S.We now give a straightforward, yet interesting property of H-line signed graphs.Theorem 2.2 For any connected graph H of order at least 3 and for any signed graph S =(G, σ), its H-line signed graph HL(S) is balanced.Proof Let σ′denote the signing ofHL(S) and let the signing σ of S be treated as a markingof the vertices of HL(S). Then by definition of HL(S) we see that σ′(e1, e2) = σ(e1)σ(e2), forevery edge (e1, e2) of HL(S) and hence, by Theorem 1.1, the result follows. Corollary 2.3 For any two signed graphs S1 and S2 with the same underlying graph, HL(S1) ∼HL(S2).The following result characterizes signed graphs which are H-line signed graphs.Theorem 2.4 A signed graph S = (G, σ) is a H-line signed graph for some connected graphH of order at least 3 if, and only if, S is balanced signed graph and its underlying graph G is a40 R.Rangarajan, M. S. Subramanya and P. Siva Kota ReddyH-line graph.Proof Suppose that S isH-line signed graph. Then there exists a signed graph S′ = (G′, σ′)such that HL(S′) ∼= S. Hence by definition HL(G) ∼= G′ and by Theorem 2.2, S is balanced.Conversely, suppose that S = (G, σ) is balanced and G is H-line graph. That is there existsa graph G′ such that HL(G′) ∼= G. Since S is balanced by Theorem 1.1, there exists a markingµ of vertices of S such that for any edge uv ∈ G, σ(uv) = µ(u)µ(v). Also since G ∼= HL(G′),vertices in G are in one-to-one correspondence with the edges of G′. Now consider the signedgraph S′ = (G′, σ′), where for any edge e′ in G′ to be the marking on the corresponding vertexin G. Then clearly HL(S′) ∼= S and so S is H-line graph. For any positive integer k, the kth iterated H-line signed graph, HLk(S) of S is defined asfollows:HL0(S) = S, HLk(S) = HL(HLk−1(S)).Corollary 2.5 Given a signed graph S = (G, σ) and any positive integer k, HLk(S) is balanced,for any connected graph H of order ≥ 3.In [6], the authors proved the following for a graph G its H-line graph HL(G) is isomorphicto G then H is a path or a cycle. Analogously we have the following.Theorem 2.6 If a signed graph S = (G, σ) satisfies S ∼ HL(S) then S is balanced and H isa cycle or a path.Theorem 2.7 For any cycle Ck on k ≥ 3 vertices, a connected graph G on n ≥ r verticessatisfies CkL(G) ∼= G if, and only if, G = Ck.Proof Suppose that CkL(G) ∼= G. Then clearly, G must be unicyclic. Since CkL(G) ∼= G,we observe that G must contain a cycle Ck. Next, suppose that G contains a vertex of degree≥ 3, then the vertex corresponding to the edge not on the cycle in CkL(G) will be isolatedvertex. Hence G must be a cycle Ck.Conversely, if G = Ck, then clearly for any two adjacent edges in Ck belongs to a copy ofCk and so CkL(G) ∼= L(G). Since the line graph of any Ck is Ck itself, we have CkL(G) ∼= G.Corollary 2.8 For any cycle Ck on k ≥ 3 vertices, a graph G on n ≥ r vertices satisfiesCkL(G) ∼= G if, and only if, G is 2-regular and every component of G is Ck.In view of the above theorem we have,Theorem 2.9 For any cycle Ck on k ≥ 3 vertices, a signed graph S = (G, σ) connected graphG on n ≥ r vertices satisfies CkL(S) ∼ S if, and only if, G = Ck.Theorem 2.10 For a path Pk on k ≥ 3 vertices a connected graph G on n ≥ r vertices whichcontains a cycle of length r > k satisfies PkL(G) ∼= L(G) if, and only if, G = Cn and n ≥ k.Proof The result follows if k = 3, since P3L(G) = L(G). Assume that k ≥ 4. ClearlyG must contain at least k vertices. Suppose that PkL(G) ∼= L(G) and G contains a cycle ofThe H-Line Signed Graph of a Signed Graph 41length r ≥ k. Then number of vertices in G and number of edges are equal. Hence G mustbe unicyclic. Since G contains a cycle of length r > k, then any two adjacent edges in C ofG belongs to a common Pk. Hence PkL(G) also contains a cycle of length r. Next, supposethat G contains a vertex of degree ≥ 3, then the vertex corresponding to the edge not on thecycle in PkL(G) will be adjacent to two adjacent vertices forming a C3 and so HL(G) is notunicyclic. Hence G must be the cycle Cn .Conversely, if G = Cn and n ≥ k, then clearly any two adjacent edges in Ck belongs to acopy of Ck and so PkL(G) ∼= L(G). Since the line graph of Cn is Cn itself, PkL(G) ∼= L(G). Corollary 2.11 For any path Pk on k ≥ 3 vertices, a graph G on n ≥ r vertices satisfiesPkL(G) ∼= G if, and only if, G is 2-regular and every component of G is Cr, for some r ≥ k.Analogously, we have the following for signed graphs:Corollary 2.12 For any path Pk on k ≥ 3 vertices, a signed graph S = (G, σ) on n ≥ r verticessatisfies PkL(S) ∼ S if, and only if, S is balanced and every component of G is Cr, for somer ≥ k.In [10], the authors prove that for any graph G, T (G) ∼= L(G) if, and only if, G = Kn.Analogously, we have the following:Theorem 2.13 A graph G of order n satisfies KrL(G) ∼= L(G) for some r ≤ n if, and only if,G = Kn.Proof The result is trivial if k = n. Suppose that KrL(G) ∼= L(G) and G is not completefor some r ≤ n − 1. Then there exists at least two nonadjacent vertices u and v in G. Nowfor any vertex w, the edges uw and vw are adjacent and hence the corresponding vertices areadjacent. But the edges uw and vw can not be adjacent in KrL(G) since any set of r verticescontaining u and v can not induce complete subgraph Kr. Whence, the condition is necessary.For sufficiency, suppose G = Kn for some n ≥ r. Then for any two adjacent verticesin L(G), the corresponding edges adjacent edges in G belongs to some Kr. Hence they arealso adjacent in KrL(G) and any two nonadjacent vertices in L(G) remain nonadjacent.Thiscompletes the proof. Analogously, we have the following result for signed graphs:Theorem 2.14 A signed graph S = (G, σ) satisfies KrL(S) ∼ L(S), for some 3 ≤ k ≤ |V (G)|if, and only if, S is a balanced on a complete graph.§3. Triangular Line Signed Graphs and (0, 1, -1) MatricesMatrices are very good models to represent a graph. In general given a matrix A = (aij) oforder m × n one can associate many graphs with it (see [11]. On the other hand given anygraph G we can associate many matrices such adjacency matric, incidence matrix etc (see [8]).Analogously, given a matrix with entries one can associate many signed graphs (See [11]). In42 R.Rangarajan, M. S. Subramanya and P. Siva Kota Reddythis section, we give a relation between the notion of triangular line graph and some graphassociated with {0, 1}-matrices. Also we extend this to triangular signed graphs and somesigned graphs associated with matrices whose entries are −1, 0, or 1.Given a (0, 1)-matrix A, the term graph T (A) of A was defined as follows (See [2]): Thevertex set of T (A) consists of m row labels r1, r2, ..., rm and n column labels c1, c2, ..., cn of Aand the edge set consists of the unordered pairs ricj for which aij 6= 0.Given a (0, 1)-matrix A of order m × n, the graph Gt(A) can be constructed as follows:The vertex set of Gt(A) consists of non-zero entries of A and the edge set consists of distinctpairs of vertices (aij , akr) that lie in the same row (i=k) with air 6= 0 or or same column(j=r)with akj 6= 0. The following result relates the connects the two notions the term graph and Gtgraph of a given matrix A:Theorem 3.1 For any (0, 1)-matrix A, Gt(A) = T (T (A)).Let A = (aij) be any m × n matrix in which each entry belongs to the set {−1, 0, 1}; weshall call such a matrix a (0,±1)-matrix. The notion of term graph of a (0, 1)-matrix can beeasily extended to term signed graph of a (0,±1)-matrix A as follows ( see [2]): The vertex setof T (A) consists of m row labels r1, r2, ..., rm and n column labels c1, c2, ..., cn of A, the edgeset consists of the unordered pairs ricj for which aij 6= 0 and the sign of the edge ricj is thesign of the nonzero entry aij .Next, given any (0,±1)-matrix A a triangular matrix signed graph Sgt(A) of A can beconstructed as follows: The vertex set of Sgt(A) is consists of nonzero entries of A and edge setconsists of distinct pairs of vertices (aij , akr) that lie in the same row (i = k) with air 6= 0 orsame column (j = r) with akj 6= 0; the sign of an edge uv in Sg(A) is defined as the product ofsings of the entries of A that correspond to u = aij and v = akr.The following is a observation whose proof follows from the definition of triangular linegraph and the facts just mentioned above:Theorem 3.2 For any (0,±1) matrix A, Sgt(A) ∼= T (Tg(A)).The Kronecker product or tensor product of two signed graphs S1 and S2, denoted byS1⊗S2 is defined (see [2]) as follows:The vertex set of (S1⊗S2) is V (S1) × V (S2), the edgeset is E(S1⊗S2) := {((u1, v1), (u2, v2)) : u1u2 ∈ E(S1), v1v2 ∈ E(S2)} and the sign of theedge (u1, v1)(u2, v2) is the product of the sign of u1u2 in S1 and the sign of v1v2 in S2. In thefollowing result, A(S) will denote the usual adjacency matrix of the given signed graph S andA⊗B denotes the standard tensor product of the given matrices A and B.Theorem 3.3( M. Acharya [2]) For any two signed graphs S1andS2, A(S1⊗S2) = A(S1)⊗A(S2).Theorem 3.4 For any signed graph S, T (A(S)) = S⊗K+2 , where K+2 denotes the completegraph K2 with its only edge treated as being positive.The adjacency signed graph ð(S) of a given signed graph S is the matrix signed graphSg(A(S)) of the adjacency matrix A(S) of S [2].The H-Line Signed Graph of a Signed Graph 43Theorem 3.5( M. Acharya [2]) For any signed graph S, ð(S) = L(S⊗K+2 ).Analogously we define triangular adjacency signed graph of A(S), the adjacency matrix ofS denoted by ðt(S) as the signed graph Sgt(A(S)). We have the following result.Theorem 3.6 For any signed graph S, ðt(S) = T (S⊗K+2 ).[1] R. P. Abelson and M. J. Rosenberg, Symoblic psychologic: A model of attitudinal cognition,Behav. Sci., 3 (1958), 1-13.[2] M. 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The H-Line Signed Graph of a Signed Graph

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