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Articles 1 - 6 of 6
Full-Text Articles in Physical Sciences and Mathematics
Classification Of Book Representations Of K6, Dana Rowland
Classification Of Book Representations Of K6, Dana Rowland
Mathematics Faculty Publications
A book representation of a graph is a particular way of embedding a graph in three dimensional space so that the vertices lie on a circle and the edges are chords on disjoint topological disks. We describe a set of operations on book representations that preserves ambient isotopy, and apply these operations to K6, the complete graph with six vertices. We prove there are exactly 59 distinct book representations for K6, and we identify the number and type of knotted and linked cycles in each representation. We show that book representations of K6 contain between one and seven links, and …
Colorings, Determinants And Alexander Polynomials For Spatial Graphs, Terry Kong, Alec Lewald, Blake Mellor, Vadim Pigrish
Colorings, Determinants And Alexander Polynomials For Spatial Graphs, Terry Kong, Alec Lewald, Blake Mellor, Vadim Pigrish
Mathematics Faculty Works
A {\em balanced} spatial graph has an integer weight on each edge, so that the directed sum of the weights at each vertex is zero. We describe the Alexander module and polynomial for balanced spatial graphs (originally due to Kinoshita \cite{ki}), and examine their behavior under some common operations on the graph. We use the Alexander module to define the determinant and p-colorings of a balanced spatial graph, and provide examples. We show that the determinant of a spatial graph determines for which p the graph is p-colorable, and that a p-coloring of a graph corresponds to a representation of …
Complete Bipartite Graphs Whose Topological Symmetry Groups Are Polyhedral, Blake Mellor
Complete Bipartite Graphs Whose Topological Symmetry Groups Are Polyhedral, Blake Mellor
Mathematics Faculty Works
We determine for which n, the complete bipartite graph Kn,n has an embedding in S3 whose topological symmetry group is isomorphic to one of the polyhedral groups: A4, A5, or S4.
Symmetries Of Embedded Complete Bipartite Graphs, Erica Flapan, Nicole Lehle '09, Blake Mellor, Matt Pittluck, Xan Vongsathorn '09
Symmetries Of Embedded Complete Bipartite Graphs, Erica Flapan, Nicole Lehle '09, Blake Mellor, Matt Pittluck, Xan Vongsathorn '09
Pomona Faculty Publications and Research
We characterize which automorphisms of an arbitrary complete bipartite graph Kn,m can be induced by a homeomorphism of some embedding of the graph in S3.
Symmetries Of Embedded Complete Bipartite Graphs, Erica Flapan, Nicole Lehle, Blake Mellor, Matt Pittluck, Xan Vongsathorn
Symmetries Of Embedded Complete Bipartite Graphs, Erica Flapan, Nicole Lehle, Blake Mellor, Matt Pittluck, Xan Vongsathorn
Mathematics Faculty Works
We characterize which automorphisms of an arbitrary complete bipartite graph Kn,m can be induced by a homeomorphism of some embedding of the graph in S3.
Counting Links And Knots In Complete Graphs, Loren Abrams, Blake Mellor, Lowell Trott
Counting Links And Knots In Complete Graphs, Loren Abrams, Blake Mellor, Lowell Trott
Mathematics Faculty Works
We investigate the minimal number of links and knots in embeddings of complete partite graphs in S3. We provide exact values or bounds on the minimal number of links for all complete partite graphs with all but 4 vertices in one partition, or with 9 vertices in total. In particular, we find that the minimal number of links in an embedding of K4,4,1 is 74. We also provide exact values or bounds on the minimal number of knots for all complete partite graphs with 8 vertices.