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The triangular line \(n\)-sigraph of a symmetric \(n\)-sigraph. (English) Zbl 1213.05120
Summary: An \(n\)-tuple \((a_1,a_2,\dots, a_n)\) is symmetric, if \(a_k= a_{n-k+1}\), \(1\leq k\leq n\). Let \[ H_n = \{(a_1,a_2,\dots, a_n): a_k\in\{+,-\},\;a_k= a_{n-k+1}, \;1\leq k\leq n\} \] be the set of all symmetric \(n\)-tuples. A symmetric \(n\)-sigraph (symmetric \(n\)-marked graph) is an ordered pair \(S_n= (G,\sigma)\) \((S_n= (G,\mu))\), where \(G= (V, E)\) is a graph called the underlying graph of \(S_n\) and \(\sigma: E\to H_n(\mu: V\to H_n)\) is a function. Analogous to the concept of the triangular line graph of a graph, the triangle line symmetric \(n\)-sigraph of a symmetric \(n\)-sigraph is defined. It is shown that for any symmetric \(n\)-sigraph \(S_n\), its triangular line symmetric \(n\)-sigraph is \(i\)-balanced. We then give structural characterization of triangular line symmetric \(n\)-sigraphs. Further, we obtain some switching equivalence relationship between triangular line symmetric \(n\)-sigraph and line symmetric \(n\)-sigraph.
05C22 Signed and weighted graphs