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MAXIMAL EXPONENTS OF PRIMITIVE GRAPHS WITH MINIMUM DEGREE 3

  • Received : 2011.09.15
  • Accepted : 2011.12.05
  • Published : 2011.12.30

Abstract

In this paper, we find the maximum exponent of primitive simple graphs G under the restriction $deg(v){\geq}3$ for all vertex $v$ of G. Our result is also an answer of a Klee and Quaife type problem on exponent to find minimum number of vertices of graphs which have fixed even exponent and the degree of whose vertices are always at least 3.

Keywords

Acknowledgement

Supported by : Gangneung-Wonju National University

References

  1. R.A. Brualdi and J.A. Ross, On the exponent of a primitive, nearly reducible matrix, Math. Oper. Res. 5(1980), 229-241. https://doi.org/10.1287/moor.5.2.229
  2. J. Cai, K.M. Zhang, The characterization of symmertric primitive matrices with exponent 2n-2r, Linear Multilinear Algebra 39(1995), 391-396. https://doi.org/10.1080/03081089508818409
  3. A. Dulmage and N. Mendelsohn, On the exponent of a primitive,nearly reducible matrix, Math. Oper. Res. 5(1980), 229-241. https://doi.org/10.1287/moor.5.2.229
  4. J.C. Holladay and R.S. Varga, On powers of non-negative matrices, Proc. Amer. Math. Soc. 9(1958), 631-634. https://doi.org/10.1090/S0002-9939-1958-0097416-8
  5. B.M. Kim, B.C. Song and W. Hwang , Wielandt type theorem for cartesian product of digraphs, Linear Algebra Appl. 429(2008), 841-848. https://doi.org/10.1016/j.laa.2008.04.029
  6. V. Klee, Classification and enumeration of minimum (d, 3, 3)-graphs for odd d, J. Combin. Theory Ser. B 28(1980), 184-207. https://doi.org/10.1016/0095-8956(80)90065-9
  7. V. Klee and H. Quaife, Minimum graphs of specified diameter, connectivity and valence, Math. Oper. Res. 1(1)(1976), 28-31. https://doi.org/10.1287/moor.1.1.28
  8. V. Klee and H. Quaife, Classification and enumeration of minimum (d, 1, 3)- graphs and minimum (d, 2, 3)-graphs, J. Combin. Theory Ser. B 23(1977), 83- 93. https://doi.org/10.1016/0095-8956(77)90059-4
  9. M. Lewin, On exponents of primitive matrices, Numer. Math. 18(1971), 154- 161. https://doi.org/10.1007/BF01436324
  10. J.W. Moon and N.J. Pullman, On the powers of tournament matrices, J. Combinatorial Theory 3(1967), 1-9. https://doi.org/10.1016/S0021-9800(67)80009-7
  11. J. Ross, On the exponent of a primitive, nearly reducible matrix. II, SIAM J. Alg. Discr. Meth. 3(1982), 395-410. https://doi.org/10.1137/0603040
  12. J. Shao, The exponent set of primitive, nearly reducible matrices, SIAM J. Alg. Discr. Meth. 8(1987), 578-584. https://doi.org/10.1137/0608047
  13. J. Shen, Exponents of 2-regular digraphs, Discrete Math. 214(2000), 211-219. https://doi.org/10.1016/S0012-365X(99)00144-2
  14. H. Wielandt, Unzerleghare, nicht negative Matrizen, Math. Z. 52(1950) 642-645. https://doi.org/10.1007/BF02230720
  15. K. Zhang, On Lewin and Vitek's conjecture about the exponent set of primitve matrices, Linear Algebra Appl. 96(1987), 101-108. https://doi.org/10.1016/0024-3795(87)90338-7