The Line n-sigraph of a Symmetric n-sigraph-V

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 of all symmetric n-tuples. Asymmetric n-sigraph (symmetric n-marked graph) is an ordered pair Sn = (G,�) (Sn = (G,µ)), where G = (V,E) is a graph called the underlying graph of Sn and � : E → Hn (µ : V → Hn) 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 Lr(G). The vertices of Lr(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 Sn = (G,�) as a symmetric n- sigraph Lr(Sn) = (Lr(G),� ' ), where Lr(G) is the underlying graph of Lr(Sn), where for any edge PQ in Lr(Sn), � ' (PQ) = �(P)�(Q). It is shown that for any symmetric n- sigraph Sn, its Lr(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 Sn ∼ L2(Sn), L2(Sn) ∼ L(Sn) and L2(Sn) ∼ Sn where ∼ denotes switching equivalence and L2(Sn), L(Sn) and Sn are denotes the super line symmetric n-sigraph of index 2, line symmetric n-sigraph and complementary symmetric n-sigraph of Sn respectively. Also, we characterize symmetric n-sigraphs Sn for which Sn ∼ L2(Sn) and L2(Sn) ∼ L(Sn).