Topic
Biggs–Smith graph
About: Biggs–Smith graph is a research topic. Over the lifetime, 4 publications have been published within this topic receiving 2 citations.
Papers
TL;DR: In this paper, the authors give four descriptions of the distance regular graph with 102 vertices, diameter 7 and girth 9 found by Biggs and Smith, and show that this graph is characterized by its intersection array.
Abstract: We give four descriptions of the distance regular graph with 102 vertices, diameter 7 and girth 9 found by Biggs and Smith, and we show that this graph is characterized by its intersection array.
2 citations
Posted Content•
2 Feb 2010TL;DR: The distance-3 graph S3 of the Biggs-Smith graph S is shown to be a connected edge-disjoint union of 102 tetrahedra (copies of K4) and as such the K4-ultrahomogeneous Menger graph of a self-dual (1024)-configuration as mentioned in this paper.
Abstract: The distance-3 graph S3 of the Biggs-Smith graph S is shown to be: (a) a connected edge-disjoint union of 102 tetrahedra (copies of K4) and as such the K4-ultrahomogeneous Menger graph of a self-dual (1024)-configuration; (b) a union of 102 cuboctahedra, (copies of L(Q3)), with no 2 such cuboctahedra having a common chordless 4-cycle; (c) not a line graph. Moreover, S3 is shown to have a C-ultrahomogeneous property for C = {K4}∪ {L(Q3)} restricted to preserving a specific edge partition of L(Q3) into 2-paths, with each triangle (resp. each edge) shared by 2 copies of L(Q3) plus one of K4 (resp. 4 copies of L(Q4)). Both the distance-2 and distance-4 graphs, S 2 and S4, of S appear in the context associated with the above mentioned edge partition. This takes us to ask whether there are any non-line-graphical connected K4-ultrahomogeneous Menger graphs of self-dual (n4)-configurations that are edge-disjoint unions of several copies of K4, for positive integers n / ∈ {42, 102}.
1 citations
Posted Content•
2 Feb 2010
TL;DR: In this article, it was shown that a self-dual 1-configurations possess their Menger graph most symmetric if $K_d$-ultrahomogeneous.
Abstract: Self-dual 1-configurations $(n_d)_1$ possess their Menger graph $\mathcal Y$ most $K_4$-separated among connected self-dual configurations $(n_d)$. Such $\mathcal Y$ is most symmetric if $K_d$-ultrahomogeneous. In this work, such a $\mathcal Y$ is presented for $(n,d)=(102,4)$ and shown to relate $n$ copies of the cuboctahedral graph $L(Q_3)$ to the $n$ copies of $K_d$; these are shown to share each copy of $K_3$ exactly with two copies of $L(Q_3)$.
Posted Content•
2 Feb 2010TL;DR: The distance-3 graph S of the Biggs-Smith graph S is shown to be both a connected edge-disjoint union of 102 tetrahedra, (copies of K4), and a union with no two such cuboctahedra having a common chordless 4-cycle as discussed by the authors.
Abstract: The distance-3 graph S of the Biggs-Smith graph S is shown to be both a connected edge-disjoint union of 102 tetrahedra, (copies of K4), and a union of 102 cuboctahedra, (copies of L(Q3)), with no two such cuboctahedra having a common chordless 4-cycle. Moreover, S is shown to have a C-ultrahomogeneous property for C = {K4} ∪ {L(Q3)} restricted to preserving a specific edge decomposition of L(Q3) into 2-paths, with each triangle (resp. edge) shared by two copies of L(Q3) plus one of K4 (resp. four copies of L(Q4)) exactly. Furthermore, S 3 is shown to be the Menger graph of a self-dual (1024)-configuration. Both the distance-2 and distance-4 graphs, S and S, of S appear in the context associated to the above mentioned edge decomposition. As a result, the connected edge disjoint union of copies of K4 (Menger graph of a self-dual (424)-configuration) forming the unique non-line-graphical K4ultrahomogeneous graph in the literature is accompanied now by S as a second such graph.
Performance Metrics
| Year | Papers |
|---|---|
| 2010 | 3 |
| 1988 | 1 |