Geodesic
The 3-path-step operator on trees and unicyclic graphs
Mathematica Bohemica, Tome 127 (2002) no. 1, pp. 33-40

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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 $S^{\prime }_k$ which to every finite graph $G$ assigns the graph $S^{\prime }_k(G)$ 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 $S^{\prime }_3$ are studied.
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 $S^{\prime }_k$ which to every finite graph $G$ assigns the graph $S^{\prime }_k(G)$ 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 $S^{\prime }_3$ are studied.
DOI : 10.21136/MB.2002.133982
Classification : 05C05, 05C38
Keywords: 3-path-step graph operator; tree; unicyclic graph 
Zelinka, Bohdan. The 3-path-step operator on trees and unicyclic graphs. Mathematica Bohemica, Tome 127 (2002) no. 1, pp. 33-40. doi: 10.21136/MB.2002.133982
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[1] F. Escalante, L. Montejano: Trees and $n$-path invariant graphs, Abstract. Graph Theory Newsletter 33 (1974).

[2] E. Prisner: Graph Dynamics. Longman House, Burnt Mill, Harlow, 1998. | MR

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