In this note we will generalize the Higman-Haemers inequalities for generalized polygons to thick regular near polygons.

The groups Gk,1 of Richard Thompson and Graham Higman can be generalized in a natural way to monoids, that we call Mk,1, and to inverse monoids, called Invk,1; this is done by simply generalizing bijections to partial functions or partial injective functions. The monoids Mk,1 have connections with circuit complexity (studied in another paper). Here we prove that Mk,1 and Invk,1 are congruence-s...

By Ulam’s conjecture every finite graph G can be reconstructed from its deck of vertex deleted subgraphs. The conjecture is still open, but many special cases have been settled. In particular, one can reconstruct Cartesian products. We consider the case of k-vertex deleted subgraphs of Cartesian products, and prove that one can decide whether a graph H is a kvertex deleted subgraph of a Cartesi...

A construction of Tits is used to cast the argument of Kilmoyer-Solomon and Higman proving the Feit-Higman theorem in the context of tactical configurations. An analog of a result of Cvetkovic shows what combinatorial data is needed to determine the representations associated with a given configuration. The possible representations fall into three types. A bound is given for the size of certain...

There does not exist an isomorphism-invariant Borel version of the Higman-Neumann-Neumann Embedding Theorem.

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The connective constant μ(G) of an infinite transitive graph G is the exponential growth rate of the number of self-avoiding walks from a given origin. The relationship between connective constants and amenability is explored in the current work. Various properties of connective constants depend on the existence of so-called ‘unimodular graph height functions’, namely: (i) whether μ(G) is a loc...