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The Exponents of Strongly Connected Graphs

  • Edward A. Bender (a1) and Thomas W. Tucker (a2)

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A directed graph G is a set of vertices V and a subset of V × V called the edges of G. A path in G of length k,

is such that (vi, vi+1) is an edge of G for 1 ≦ ik. A directed graph G is strongly connected if there is a path from every vertex of G to every other vertex. A circuit is a path whose two end vertices are equal. An elementary circuit has no other equal vertices. See (1) for a fuller discussion.

Let G be a finite, strongly connected, directed graph (fscdg). The kth power Gk of G is the directed graph with the same vertices as G and edges of the form (i, j) where G has a path of length k from i to j.

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References

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1. Berge, C., The theory of graphs and its applications (Wiley, New York, 1962).
2. Brauer, A. and Schockley, J. E., On a problem of Frobenius, J. Reine Angew. Math. 211 (1962), 215220.
3. Dulmage, A. L. and Mendelsohn, N. S., Gaps in the exponent set of primitive matrices, Illinois J. Math. 8 (1964), 642656.
4. Heap, B. R. and Lynn, M. S., The structure of powers of non-negative matrices. I. The index of convergence (National Physical Laboratory, Teddington, Middlesex, England, 1964), SIAM J. Appl. Math. 14 (1966), 610639.
5. Norman, R. Z., private communication.
6. Ptâk, V. and Sedlâcek, J., On the index of imprimitivity of non-negative matrices, Czechoslovak Math. J. 8 (83) (1958), 496501.
7. Roberts, J. B., Note on linear forms, Proc. Amer. Math. Soc. 7 (1956), 465469.
8. Tucker, T. W., The exponent set of strongly connected graphs, Senior thesis, Harvard University, 1967.
9. Wielandt, H., Unzerlegbare, nicht negativen Matrizen, Math. Z. 52 (1950), 642648.
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