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MATHEMATICA BOHEMICA, Vol. 127, No. 1, pp. 33-40 (2002)
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# The 3-path-step operator on trees

and unicyclic graphs

## Bohdan Zelinka

* Bohdan Zelinka*, Department of Applied Mathematics, Technical University of Liberec, Voronezska 13, 460 01 Liberec 1, e-mail: ` bohdan.zelinka@vslib.cz`

**Abstract:**
E. Prisner in his book Graph Dynamics 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.

**Keywords:** 3-path-step graph operator, tree, unicyclic graph

**Classification (MSC2000):** 05C38, 05C05

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