On Narrow Hexagonal Graphs with a 3-Homogeneous Suborbit
Manley Perkel
, Cheryl E. Praeger
and Richard Weiss
DOI: 10.1023/A:1011208230870
Abstract
A connected graph of girth m 
3 is called a polygonal graph if it contains a set of m-gons such that every path of length two is contained in a unique element of the set. In this paper we investigate polygonal graphs of girth 6 or more having automorphism groups which are transitive on the vertices and such that the vertex stabilizers are 3-homogeneous on adjacent vertices. We previously showed that the study of such graphs divides naturally into a number of substantial subcases. Here we analyze one of these cases and characterize the k-valent polygonal graphs of girth 6 which have automorphism groups transitive on vertices, which preserve the set of special hexagons, and which have a suborbit of size k - 1 at distance three from a given vertex.
3 is called a polygonal graph if it contains a set of m-gons such that every path of length two is contained in a unique element of the set. In this paper we investigate polygonal graphs of girth 6 or more having automorphism groups which are transitive on the vertices and such that the vertex stabilizers are 3-homogeneous on adjacent vertices. We previously showed that the study of such graphs divides naturally into a number of substantial subcases. Here we analyze one of these cases and characterize the k-valent polygonal graphs of girth 6 which have automorphism groups transitive on vertices, which preserve the set of special hexagons, and which have a suborbit of size k - 1 at distance three from a given vertex.Pages: 257–273
Keywords: polygonal graph; automorphism group; 3-homogeneous suborbit
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References
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1994. Also at http://www.maths. usyd.edu.au:8000/comp/magma/Overview.html.
2. D.G. Higman, “Intersection matrices for finite permutation groups,” J. Alg. 6 (1967), 22-42.
3. W.M. Kantor, “k-Homogeneous groups,” Math. Z. 124 (1972), 261-265.
4. M. Perkel, “Bounding the valency of polygonal graphs with odd girth,” Can. J. Math. 31 (1979), 1307-1321.
5. M. Perkel, “Near-polygonal graphs,” Ars. Comb. 26A (1988), 149-170.
6. M. Perkel, “An infinite family of narrow hexagonal graphs with solvable 2-transitive point stabilizer,” in preparation.
7. M. Perkel and C.E. Praeger, “On narrow 6-gonal graphs with a triply transitive suborbit,” Research Report 27, The University of Western Australia, October 1996.
8. M. Perkel and C.E. Praeger, “Polygonal graphs: New families and an approach to their analysis,” Congressus Numerantium 124 (1997), 161-173.