The neighbourhood of two adjacent vertices in a Conway 99-graph

Details of a search for the possible subgraphs of a Conway 99-graph consisting of two adjacent vertices and their respective neighbours.

Constraints on Conway’s 99-graph (and its subgraphs)

A look at some results that would have to be satisfied by a Conway 99-graph, if it exists.

Conway’s 99-graph (and related problems)

An introduction to strongly regular graphs via Conway’s 99-graph problem and some of its friends.

My Erdős number…

…appears to be four (an infinite improvement). I coauthored a paper with Gary Greaves, whose recent paper Edge-signed graphs with smallest eigenvalue greater than -2 also saw contributions from Jack Koolen and Akhiro Munemasa. They both have an Erdős number of two (each via Chris Godsil, who is an Erdős coauthor), making Gary a three …

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What is a buckyball? Part 3: Fullerenes

In the previous post we saw how we could project polyhedra into the plane, and use some simple properties about planar graphs to classify all the possible Platonic solids. In this post we’ll finally get to the buckyball, by considering a less restrictive class of polyhedra: the fullerenes. The Platonic solids were extremely regular: every …

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What is a Buckyball? Part 2: Projection

How can we represent a 3-dimensional object such a cube in only 2-dimensions, such as on a flat piece of paper? This is the problem of projection, and it inevitably introduces inaccuracies. Different choices of perspective will alter what features survive the projection process. For instance, a perfect cube has all faces square, with corner …

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What is a Buckyball? Part 1: Planar Graphs

Earlier this year I was involved with the construction of the `Giant 4D Buckyball‘, as part of the University of Edinburgh’s Innovative Learning Week. The sculpture was actually of something rather more complicated – the Cantitruncated 600-cell – but buckyballs (in various representations) were a fundamental building block. So, what exactly are they? Julia’s description …

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The benign dictatorship of the London Underground

Earlier today I spotted this video, featuring the stand-up mathematician Matt Parker and all-round interesting person Tom Scott exploring some oddities of the tube: Matt’s ability to beat Tom around the network depended on local knowledge of hidden shortcuts. You might wonder why these quicker options aren’t indicated by signs, but as they explain in …

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Lehmer’s Conjecture for Hermitian Matrices over the Eisenstein and Gaussian Integers

My third paper on the Mahler measure problem has been accepted by the Electronic Journal of Combinatorics, and is available here freely under the E-JC’s open access policy. This is joint work with Gary Greaves, and completes the proof of Lehmer’s conjecture for matrices with entries from rings of integers of quadratic extensions: a project …

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