Geodesic
Graphs with Maximal Even Girth
Canadian journal of mathematics, Tome 21 (1969) no. 1, pp. 915-934

Voir la notice de l'article provenant de la source Cambridge University Press

In this paper we examine the class G of simple undirected, connected graphs of diameter d > 1, girth 2d, and for any g ɛ G, if a pair of nodes are at distance d from each other, then that pair of nodes is connected by t distinct paths of length d, t > 1. (The girth of g is the length of the smallest circuit in g.)We establish, in § 2, that for all g ɛ G, g is regular.We establish necessary conditions for the existence of elements of G. If g ɛ G, we adopt the notation g = g(d, t, v, n), where v is the valence of g and n is the number of nodes. It is of course possible for g, h ɛ G, g ≠ h, and for given d, t, v, n to have both g(d, t, v, n) and h(d, t, v, n).In particular, we show that if d = 2, t ≠ 2, 4 or 6, then there is at most a finite number of graphs with a particular given t value. 
Gewirtz, A. Graphs with Maximal Even Girth. Canadian journal of mathematics, Tome 21 (1969) no. 1, pp. 915-934. doi: 10.4153/CJM-1969-101-9
@article{10_4153_CJM_1969_101_9,
     author = {Gewirtz, A.},
     title = {Graphs with {Maximal} {Even} {Girth}},
     journal = {Canadian journal of mathematics},
     pages = {915--934},
     year = {1969},
     volume = {21},
     number = {1},
     doi = {10.4153/CJM-1969-101-9},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1969-101-9/}
}
TY  - JOUR
AU  - Gewirtz, A.
TI  - Graphs with Maximal Even Girth
JO  - Canadian journal of mathematics
PY  - 1969
SP  - 915
EP  - 934
VL  - 21
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-1969-101-9/
DO  - 10.4153/CJM-1969-101-9
ID  - 10_4153_CJM_1969_101_9
ER  - 
%0 Journal Article
%A Gewirtz, A.
%T Graphs with Maximal Even Girth
%J Canadian journal of mathematics
%D 1969
%P 915-934
%V 21
%N 1
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-1969-101-9/
%R 10.4153/CJM-1969-101-9
%F 10_4153_CJM_1969_101_9

[1] 1. Elspas, B., Goldberg, J., Short, R. A., and Stone, H. S., Investigation of propagation-limited computer networks, Stanford Research Institute Report, July, 1965. Google Scholar

[2] 2. Feit, W. and Higman, G., The non-existence of certain generalized polygons, J. Algebra 1 (1964), 114–131. Google Scholar

[3] 3. Gewirtz, A., The uniqueness of g(2, 2, 10, 56), Trans. New York Acad. Sci. 81 (1969), 656–675 Google Scholar

[4] 4. Hanani, H., The existence and construction of balanced incomplete block designs, Ann. Math. Statist. 32 (1961), 361–386. Google Scholar

[5] 5. Higman, D. and Sims, C., A simple group of order 44,352,000, Math. Z. 105 (1968), 110–113 Google Scholar

[6] 6. Hoffman, A. J., On the polynomial of a graph, Amer. Math. Monthly 70 (1963), 30–36. Google Scholar

[7] 7. Hoffman, A. J. and R. Singleton, On Moore graphs with diameters 2 and 3, IBM J. Res. 4 (1960), 497–504 Google Scholar

[8] 8. Marcus, M. and Mine, H., A survey of matrix theory and matrix inequalities (Macmillan, New York, 1964). Google Scholar

[9] 9. Ore, O., Theory of graphs, Amer. Math. Soc. Colloq. Publ., Vol. 38 (Amer. Math. Soc, Providence, R.I., 1962). Google Scholar

[10] 10. Ryser, H., Combinatorial mathematics, The Carus Mathematical Monographs, No. 14 (The Mathematical Association of America, 1963; distributed by Wiley, New York). Google Scholar

[11] 11. Singleton, R., On minimal graphs of maximal even girth, Unpublished doctoral thesis, Princeton University, Princeton, N.J., 1963. Google Scholar

[12] 12. Singleton, R., On minimal graphs of maximal even girth, J. Combinatorial Theory 1 (1966), 306–322. Google Scholar

[13] 13. Witt, E., Uber Sternersche Système, Abh. Math. Sem. Hamburg 12 (1937), 265–275. Google Scholar

Cité par Sources :