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# The dynamics of the line and path graph operators

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## Abstract

For any integerk e 1 thek- path graph Pk (G) of a graph G has all length-k subpaths ofG as vertices, and two such vertices are adjacent whenever their union (as subgraphs ofG) forms a path or cycle withk + 1 edges. Fork = 1 we get the well-known line graphP 1 (G) =L(G). Iteratedk-path graphs Pt k(G) are defined as usual by Pt k (G) := Pk(P t−1 k(G)) ift < 1, and by P1 k(G): = Pk(G). A graph G isP k -periodic if it is isomorphic to some iteratedk-path graph of itself; itP k -converges if some iteratedk-path graph of G isP k -periodic. A graph has infiniteP k -depth if for any positive integert there is a graphH such that Pt k(H) ≃G. In this paperP k -periodic if it is isomorphic to some iteratedk-path graph of itself; itP k -converges if some iteratedk-path graph of G isP k -periodic graphs,P k -periodic if it is isomorphic to some iteratedk-path graph of itself; itP k -converges if some iteratedk-path graph of G isP k -convergent graphs, and graphs with infiniteP k -periodic if it is isomorphic to some iteratedk-path graph of itself; itP k -converges if some iteratedk-path graph of G isP k -depth are characterized inside some subclasses of the class of locally finite graphs fork = 1, 2.

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## References

1. 1.

Beineke, L.W.: Characterizations of derived graphs. J. Comb. Theory9, 129–135 (1970)

2. 2.

Broersma, H.J., Hoede, C.: Path graphs. J. Graph Theory13, 427–444 (1989)

3. 3.

Ghirlanda, A.M.: Osservazioni sulle caratteristiche dei graft o singrammi. Ann. Univ. Ferrara Nuova Ser., Sez. VII11, 93–106 (1962-65)

4. 4.

Ghirlanda, A.M.: Sui graft ftniti autocommutabili. Boll. Unione Mat. Ital. III Ser.18, 281–284(1963)

5. 5.

Jung, H.A.: Zu einem Isomorphiesatz von H. Whitney für Graphen. Math. Ann.164, 270–271 (1966)

6. 6.

Menon, V.V.: The isomorphism between graphs and their adjoint graphs. Can. Math. Bull.8, 7–15 (1965)

7. 7.

Menon, V.V.: On repeated interchange graphs. Amer. Math. Monthly73, 986–989 (1966)

8. 8.

Menon, V.V.: On repeated interchange graphs II. J. Comb. Theory Ser. B11, 54–57 (1971)

9. 9.

Ore, O.: Theory of graphs: American Mathematics Society Providence, Rhode Island 1962

10. 10.

Porcu, L.: Sui graft autocommutati. Inst. Lombardo Acad. Sci. Lett. Rend. A100, 665–677 (1966)

11. 11.

Prisner, E.: Iterated graph-valued functions. Preprint TU Berlin No. 232 1989

12. 12.

Prisner, E.: Graph dynamics. Monograph in preparation.

13. 13.

Sabidussi, G.: Existenz and Struktur selbstadjungierter Graphen Beitäge zur Graphen- theorie, Beiträge zur Graphentheorie, (H. Sachs, H.-J. Voß, H. Walther ed.) pp. 121–125. Teubner, Leipzig 1968

14. 14.

Sabidussi, G.: Existence and structure of self-adjoint graphs. Math. Zeitschrift104, 257–280 (1968)

15. 15.

Schwartz, B.L.: On interchange graphs. Pacific J. Math.27, 393–396 (1968)

16. 16.

Schwartz, B.L.: Infinite self-interchange graphs. Pacific J. Math.31, 497–504 (1969)

17. 17.

Schwartz, B.L., Beineke, L.W.: Locally infinite self-interchange graphs. Proc. Amer. Math. Soc.27, 8–12 (1971)

18. 18.

Whitney, H.: Congruent graphs and the connectivity of graphs. Amer. J. Math.54, 150–168 (1932)

## Author information

Correspondence to Erich Prisner.

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Prisner, E. The dynamics of the line and path graph operators. Graphs and Combinatorics 9, 335–352 (1993). https://doi.org/10.1007/BF02988321