Paul Weichsel

• PhD
• University of Illinois Urbana-Champaign

We consider some quadratic variations of the recursive sequence introduced by Lothar Collatz in 1937.

A perfectdominatingset S of a graph Γ is a set of vertices of Γ such that every vertex of Γ is either in S or is adjacent to exactly one vertex of S. We show that a perfect dominating set of the n-cube Qn induces a subgraph of Qn whose components are isomorphic to hypercubes. We conjecture that each of these hypercubes has the same dimension. We th...

Our aim is to exhibit a short algebraic proof for the Erdös-Ko-Rado theorem. First, we summarize the method of linearly independent polynomials showing that if X 1,..., X m ⊂ [n] are sets such that X i does not satisfy any of the set of s intersection conditions R i but X i satisfies at least one condition in Rj for all j > i then $ m \leqslan...

Kaplansky introduced several classes of central simple Lie algebras in characteristic 2. We view these algebras in terms of graphs, and we classify them using a theorem of Shult characterizing graphs with the "cotriangle condition"; there is also a connection with Fischer′s theorem on groups generated by 3-transpositions. Uniqueness of these algebr...

We examine the class of distance regular graphs which can be embedded in a cube. We show that in the case of isometric embedding they are precisely the cubes, the even cycles and the ‘revolving doors’—a ‘revolving door’ is the subgraph of an odd dimensional cube whose vertices are as evenly balanced in the number of 1's and 0's as possible. In the...

We consider distance-regular graphs with adjacency matrix in 2 × 2 block form. In Theorem 1 the matrix takes the form , and we examine the cases (1) ƒ(C) = C, (2) , and (3) ƒ(x) is a polynomial of degree 2. In Theorem 2 we consider the adjacency matrix , where R takes on the values (1) R=C, (2) R = J, (3) R = J−C. In each case we obtain a character...

Let G be a simple graph with adjacency matrix A, and p(x) a polynomial with rational coefficients. If p(A) is the adjacency matrix of a graph, we denote that graph by p(G). We consider the question: Given a graph G, which polynomials p(x) give rise to a graph p(G) and what are those graphs? We give a complete answer if G is a distance-regular graph...

If A is the adjacency matrix of a graph G, then Ai is the adjacency matrix of the graph on the same vertex set in which a pair of vertices is adjacent if and only if their distance apart is i in G. If G is distance-regular, then Ai is a polynomial of degree i in A. It is shown that the converse is also true. If Ai is a polynomial in A, not necessar...

The consequences of the law ( G a (1) , hellip, G a(n) ) = 1 in a basic p -group are examined. The principal tools are a combinatorial analysis of the lattice of n -tuples of positive integers and a theorem of the author about higher-commutator subgroups. Subject classification ( Amer. Math. Soc. ( MOS ) 1970 ): 20 D 15.

Stewart [1] has given a complete classification of varieties of center-extended-by-metabelian groups of exponent p and class at most p − 1 . We will use this classification as a setting for some general results about join-irreducible varieties of p-groups and to motivate the title of this note. Full proofs will appear elsewhere.

The classification of groups according to the varieties they generate requires the study of a class of indecomposable elements. Such a class is the class of basic groups which have been studied in [4], [5] and [6]. A group is called basic if it is indecomposable qua group; that is, it is critical and indecomposable qua variety; that is, its variety...

It is well known that in the category of all groups and homomorphisms, every epimorphism is onto. This result does not hold for certain other categories of groups. The condition that an epimorphism θ: G → A is not onto is equivalent to the condition that θ( G ) is a proper subgroup A with the property that any two homomorphisms α, β on A which agre...

We investigate the structure of finite irregularp-groupsG, such that all proper subvarieties of varG contain only regularp-groups.

We describe a partition of the points of a graph which is related to its automorphism group. We then prove that the group of a tree is trivial if and only if this partition is the trivial one, and we formulate an algorithm which produces such a partition. Some application to graphs in general are also considered.

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In this paper we investigate finite metabelian p-groups from the point of view of varieties and identical relations. In other words we shall be interested in those properties of such groups which are preserved under the operations of fonning direct products, taking subgroups and taking factor groups. Ideally one would like to characterize such prop...

NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. Let [...] be a set of finite groups and define [...] to be the intersection of all sets of groups which contain [...] and are closed under the operations of subgroup, factor group and direct product. The equivalence relation defined by...