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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.
##### MSC:
 05C22 Signed and weighted graphs