The Line n-Sigraph of a Symmetric n-Sigraph-II
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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 of all symmetric n-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. Given a connected graph H of order at least 3, the H-Line Graph of a graph G = (V, E), denoted by HL(G), is a graph with the vertex set E, the edge set of G where two vertices in HL(G) are adjacent if, and only if, the corresponding 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 symmetric n-sigraph HL(Sn) of a symmetric n-sigraph Sn = (G, σ) as a symmetric n-sigraph, HL(Sn) = (HL(G), σ'),and for any edge e1e2 in HL(Sn), σ'(e1e2) = σ(e1)σ(e2). In this paper,we characterize symmetric n-sigraphs Sn which are H-line symmetric n-sigraphs and study some properties of H-line graphs as well as H-line symmetric n-sigraphs. The notion HL(Sn) which generalizes the notion of line symmetric n-sigraph L(Sn) introduced by E. Sampathkumar et al. (2010).
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. 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. Given a connected graph H of order at least 3, the H-Line Graph of a graph G = (V, E), denoted by HL(G), is a graph with the vertex set E, the edge set of G where two vertices in HL(G) are adjacent if, and only if, the corresponding 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 symmetric n-sigraph HL(Sn) of a symmetric n-sigraph Sn = (G, σ) as a symmetric n-sigraph, HL(Sn) = (HL(G), σ'),and for any edge e1e2 in HL(Sn), σ'(e1e2) = σ(e1)σ(e2). In this paper,we characterize symmetric n-sigraphs Sn which are H-line symmetric n-sigraphs and study some properties of H-line graphs as well as H-line symmetric n-sigraphs. The notion HL(Sn) which generalizes the notion of line symmetric n-sigraph L(Sn) introduced by E. Sampathkumar et al. (2010).
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