The Line n Sigraph of a Symmetric n Sigraph IV

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					International J.Math. Combin. Vol.1(2012), 106-112



                    The Line n-Sigraph of a Symmetric n-Sigraph-IV

                         P. Siva Kota Reddy† , K. M. Nagaraja‡ and M
				
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Description: An n-tuple (a^sub 1^, a^sub 2^, ... , a^sub n^) is symmetric, if a^sub n-k+1^, 1 ≤ k ≤ n. Let H^sub n^ = (a^sub 1^, a^sub 2^, ... , a^sub n^) : a^sub k^ ∈ +, - }, a^sub k^ = a^sub n-k+1^, 1 ≤ k ≤ n} be the set of all symmetric n-tuples. Asymmetric n-sigraph (symmetric n-marked graph) is an ordered pair S^sub n^ = (G, σ) (S^sub n^ = (G, μ)), where G = (V, E) is a graph called the underlying graph of S^sub n^ and σ : E [arrow right] H^sub n^ (μ : V [arrow right] H^sub n^) is a function. In Bagga et al. (1995) introduced the concept of the super line graph of index r of a graph G, denoted by L^sub r^(G). The vertices of L^sub r^(G) are the r-subsets of E(G) and two vertices P and Q are adjacent if there exist p ∈ P and q ∈ Q such that p and q are adjacent edges in G. Analogously, one can define the super line symmetric n-sigraph of index r of a symmetric n-sigraph S^sub n^ = (G, σ) as a symmetric n-sigraph L^sub r^(S^sub n^) = (L^sub r^(G), σ'), where L^sub r^(G) is the underlying graph of L^sub r^(S^sub n^), where for any edge PQ in L^sub r^(S^sub n^), σ'(PQ) = σ(P)σ(Q). It is shown that for any symmetric n-sigraph S^sub n^, its L^sub r^(S^sub n^) is i-balanced and we offer a structural characterization of super line symmetric n-sigraphs of index r. Further, we characterize symmetric n-sigraphs S^sub n^ for which S^sub n^ ~ L^sub 2^(S^sub n^) ~ L^sub 2^(S^sub n^) ~ L(S^sub n^) and L^sub 2^(S^sub n^) ~ ... where ~ denotes switching equivalence and L^sub 2^(S^sub n^), L(S^sub n^) and ... are denotes the super line symmetric n-sigraph of index 2, line symmetric n-sigraph and complementary symmetric n-sigraph of S^sub n^ respectively. Also, we characterize symmetric n-sigraphs S^sub n^ for which S^sub n^ [congruent with] L^sub 2^(S^sub n^) and L^sub 2^(S^sub n^) [congruent with] L(S^sub n^). [PUBLICATION ABSTRACT]
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