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The 3-path-step operator on trees and unicyclic graphs. (English) Zbl 0995.05076
Summary: E. Prisner in his book [Graph dynamics (Pitman Research Notes in Mathematics Series. 338. Harlow, Essex: Longman Group)(1995; Zbl 0848.05001)] defines the $$k$$-path-step operator on the class of finite graphs. The $$k$$-path-step operator (for a positive integer $$k$$) is the operator $$S'_k$$ which to every finite graph $$G$$ assigns the graph $$S'_k(G)$$ which has the same vertex set as $$G$$ and in which two vertices are adjacent if and only if there exists a path of length $$k$$ in $$G$$ connecting them. In the paper the trees and the unicyclic graphs fixed in the operator $$S'_3$$ are studied.

##### MSC:
 05C38 Paths and cycles 05C05 Trees
##### Keywords:
3-path-step graph operator; tree; unicyclic graph
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