Locally finite self-interchange graphs

Schwartz, Benjamin L.; Beineke, Lowell W.

doi:10.1090/S0002-9939-1971-0289349-5

Locally finite self-interchange graphs
HTML articles powered by AMS MathViewer

by Benjamin L. Schwartz and Lowell W. Beineke PDF

Proc. Amer. Math. Soc. 27 (1971), 8-12 Request permission

Abstract:

Graphs isomorphic to their interchanges are studied. Using prior results of more special cases, plus one new concept, it is possible to characterize all locally finite self-interchange graphs, finite and infinite, connected and disconnected, with loops and parallel edges admitted. All solutions are shown to be component-unions of graphs from six easily described classes.

References

Antonio Andreatta, Sui singrammi finiti commutati di altri, Ist. Lombardo Accad. Sci. Lett. Rend. A 98 (1964), 133–156 (Italian). MR 175804
Gary Chartrand, The existence of complete cycles in repeated line-graphs, Bull. Amer. Math. Soc. 71 (1965), 668–670. MR 175806, DOI 10.1090/S0002-9904-1965-11389-1
Anna Maria Ghirlanda, Osservazioni sulle caratteristiche dei grafi o singrammi, Ann. Univ. Ferrara Sez. VII (N.S.) 11 (1962/65), 93–106 (Italian, with French summary). MR 190030
J. Krausz, Démonstration nouvelle d’une théorème de Whitney sur les réseaux, Mat. Fiz. Lapok 50 (1943), 75–85 (Hungarian, with French summary). MR 18403
Livio Porcu, Sui grafi autocommutati, Ist. Lombardo Accad. Sci. Lett. Rend. A 100 (1966), 665–677 (Italian). MR 207589
V. V. Menon, The isomorphism between graphs and their adjoint graphs, Canad. Math. Bull. 8 (1965), 7–15. MR 175114, DOI 10.4153/CMB-1965-002-7
V. V. Menon, On repeated interchange graphs, Amer. Math. Monthly 73 (1966), 986–989. MR 202631, DOI 10.2307/2314503
V. V. Menon, Repeated adjoints of graphs, Theory of Graphs (Internat. Sympos., Rome, 1966) Gordon and Breach, New York; Dunod, Paris, 1967, pp. 245–248 (English, with French summary). MR 0223266
Luigi Muracchini and Anna Maria Ghirlanda, Sul grafo commutato e sul grafo opposto di un grafo orientato, Atti Sem. Mat. Fis. Univ. Modena 14 (1965), 87–97 (Italian). MR 0194360
Oystein Ore, Theory of graphs, American Mathematical Society Colloquium Publications, Vol. XXXVIII, American Mathematical Society, Providence, R.I., 1962. MR 0150753
A. C. M. van Rooij and H. S. Wilf, The interchange graph of a finite graph, Acta Math. Acad. Sci. Hungar. 16 (1965), 263–269 (English, with Russian summary). MR 195761, DOI 10.1007/BF01904834
Gert Sabidussi, Existence and structure of self-adjoint graphs, Math. Z. 104 (1968), 257–280. MR 227044, DOI 10.1007/BF01110333
Benjamin L. Schwartz, On interchange graphs, Pacific J. Math. 27 (1968), 393–396. MR 234860
B. L. Schwartz, Infinite self-interchange graphs, Pacific J. Math. 31 (1969), 497–504. MR 262107
B. L. Schwartz, Defining the Interchange of a Loop, Math. Mag. 43 (1970), no. 3, 141–143. MR 1571864
Hassler Whitney, Congruent Graphs and the Connectivity of Graphs, Amer. J. Math. 54 (1932), no. 1, 150–168. MR 1506881, DOI 10.2307/2371086