Author
Colin Starr
Bio: Colin Starr is an academic researcher from Willamette University. The author has contributed to research in topics: Twin prime & Prime power. The author has an hindex of 4, co-authored 12 publications receiving 37 citations.
Topics: Twin prime, Prime power, Odd graph, Delannoy number, Matrix polynomial
Papers
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TL;DR: In this paper, the authors present a characterization of abelian groups with this property and resolve the following problem: if the subgroup lattice is required to be drawn hierarchically (that is, in monotonic order of index within the group), when is it possible to draw the lattice without crossings?
Abstract: In abstract algebra courses, teachers are often confronted with the task of drawing subgroup lattices. For purposes of instruction, it is usually desirable that these lattices be planar graphs (with no crossings). We present a characterization of abelian groups with this property. We also resolve the following problem in the abelian case: if the subgroup lattice is required to be drawn hierarchically (that is, in monotonic order of index within the group), when is it possible to draw the lattice without crossings?
17 citations
9 citations
TL;DR: It is proved that trees, cycles, and bipartite graphs are prime distance graphs, and that Dutch windmill graphs and paper mill graphs arePrime distance graphs if and only if the Twin Prime Conjecture and dePolignac’s Conjectures are true, respectively.
8 citations
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TL;DR: In this article, the polynomial integrand of an integral formula that yields the expected length of the minimal spanning tree of a graph whose edges are uniformly distributed over the interval is investigated.
Abstract: In this paper, we investigate the polynomial integrand of an integral formula that yields the expected length of the minimal spanning tree of a graph whose edges are uniformly distributed over the interval [0; 1] In particular, we derive a general formula for the coecients of the polynomial and apply it to express the rst few coecients in terms of the structure of the underlying graph; eg number of vertices, edges and cycles
4 citations
TL;DR: A novel derivation of the generating functions for these numbers in arbitrary dimension is given and a new proof of a recent result relating the total number of chains to the central Delannoy numbers is given.
3 citations
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TL;DR: In this paper, Nordhaus and Gaddum gave lower and upper bounds on the sum and product of the chromatic number of a graph and its complement, in terms of the order of the graph.
198 citations
TL;DR: In this paper, the mapping class group of an infinite-type surface admits an action with unbounded orbits on a connected graph whose vertices are simple closed curves on the surface.
Abstract: We study when the mapping class group of an infinite-type surface $S$ admits an action with unbounded orbits on a connected graph whose vertices are simple closed curves on $S$. We introduce a topological invariant for infinite-type surfaces that determines in many cases whether there is such an action. This allows us to conclude that, as non-locally compact topological groups, many big mapping class groups have nontrivial coarse geometry in the sense of Rosendal.
37 citations
TL;DR: In this article, it was shown that no nonabelian group whose order has three distinct prime factors can be planar, i.e., it cannot have a planar subgroup graph.
Abstract: It is natural to ask when a group has a planar Hasse lattice or more generally when its subgroup graph is planar. In this paper, we completely answer this question for finite groups. We analyze abelian groups, p-groups, solvable groups, and nonsolvable groups in turn. We find seven infinite families (four depending on two parameters, one on three, two on four), and three "sporadic" groups. In particular, we show that no nonabelian group whose order has three distinct prime factors can be planar.
28 citations