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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.

05C38 Paths and cycles
05C05 Trees
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