The Line n-sigraph of a Symmetric n-sigraph-V
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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 ofall symmetric n-tuples. A symmetric n-sigraph (symmetric n-marked graph) is anordered pair Sn = (G, σ) (S_n = (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. The restricted super line graph of index r} of a graph G, denoted by RLr(G). The vertices of RLr(G) are the r-subsets of E(G) and two vertices P = {p1, p2,...,pr} and Q = {q1, q2,...,qr} are adjacent if there exists exactly one pair of edges, say pi and qj, where 1 ≤ i,j ≤ r, that are adjacent edges in G. Analogously, one can define the restricted super line symmetric n-sigraph of index r of a symmetric n-sigraph Sn = (G, σ) as a symmetric n-sigraph RLr(Sn) = (RLr(G), σ'), where RLr(G) is the underlying graph of RLr(Sn), where for any edge PQ in RLr(Sn), σ'(PQ) = σ(P)σ(Q). It is shown that for any symmetric n-sigraph Sn, its RLr(Sn) is i-balanced and we offer a structural characterization of super line symmetric n-sigraphs of index r. Further, we characterize symmetric n-sigraphs Sn for which RLr(Sn) ~ Lr(Sn) and RLr(Sn) =~ Lr(Sn), where ~ and=~ denotes switching equivalence and isomorphism and RLr(Sn) and Lr(Sn) are denotes therestricted super line symmetric n-sigraph of index r and super linesymmetric n-sigraph of index r of Sn respectively.
Abstract
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 ofall symmetric n-tuples. A symmetric n-sigraph (symmetric n-marked graph) is anordered pair Sn = (G, σ) (S_n = (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. The restricted super line graph of index r} of a graph G, denoted by RLr(G). The vertices of RLr(G) are the r-subsets of E(G) and two vertices P = {p1, p2,...,pr} and Q = {q1, q2,...,qr} are adjacent if there exists exactly one pair of edges, say pi and qj, where 1 ≤ i,j ≤ r, that are adjacent edges in G. Analogously, one can define the restricted super line symmetric n-sigraph of index r of a symmetric n-sigraph Sn = (G, σ) as a symmetric n-sigraph RLr(Sn) = (RLr(G), σ'), where RLr(G) is the underlying graph of RLr(Sn), where for any edge PQ in RLr(Sn), σ'(PQ) = σ(P)σ(Q). It is shown that for any symmetric n-sigraph Sn, its RLr(Sn) is i-balanced and we offer a structural characterization of super line symmetric n-sigraphs of index r. Further, we characterize symmetric n-sigraphs Sn for which RLr(Sn) ~ Lr(Sn) and RLr(Sn) =~ Lr(Sn), where ~ and=~ denotes switching equivalence and isomorphism and RLr(Sn) and Lr(Sn) are denotes therestricted super line symmetric n-sigraph of index r and super linesymmetric n-sigraph of index r of Sn respectively.
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