Papers by James Cruickshank
Communications in Algebra, 2018
We classify all non-degenerate skew-hermitian forms defined over certain local rings, not necessa... more We classify all non-degenerate skew-hermitian forms defined over certain local rings, not necessarily commutative, and study some of the fundamental properties of the associated unitary groups, including their orders when the ring in question is finite.
Journal of Algebra, 2020
Let B be a ring, not necessarily commutative, having an involution * and let U 2m (B) be the unit... more Let B be a ring, not necessarily commutative, having an involution * and let U 2m (B) be the unitary group of rank 2m associated to a hermitian or skew hermitian form relative to *. When B is finite, we construct a Weil representation of U 2m (B) via Heisenberg groups and find its explicit matrix form on the Bruhat elements. As a consequence, we derive information on generalized Gauss sums. On the other hand, there is an axiomatic method to define a Weil representation of U 2m (B), and we compare the two Weil representations thus obtained under fairly general hypotheses. When B is local, not necessarily finite, we compute the index of the subgroup of U 2m (B) generated by its Bruhat elements. Besides the independent interest, this subgroup and index are involved in the foregoing comparison of Weil representations.
We investigate properties of sparse and tight surface graphs. In particular we derive topological... more We investigate properties of sparse and tight surface graphs. In particular we derive topological inductive constructions for (2, 2)-tight surface graphs in the case of the sphere, the plane, the twice punctured sphere and the torus. In the case of the torus we identify all 116 irreducible base graphs and provide a geometric application involving contact graphs of configurations of circular arcs.
Proceedings of the London Mathematical Society, 2018
A simple graph is 3-rigid if its generic embeddings in R 3 are infinitesimally rigid. Necessary a... more A simple graph is 3-rigid if its generic embeddings in R 3 are infinitesimally rigid. Necessary and sufficient conditions are obtained for the minimal 3-rigidity of a simple graph obtained from a triangulated torus by the deletion of edges interior to an embedded triangulated disc.
arXiv: Combinatorics, 2020
We give a short proof of a result of Jordan and Tanigawa that a 4-connected graph which has a spa... more We give a short proof of a result of Jordan and Tanigawa that a 4-connected graph which has a spanning planar triangulation as a proper subgraph is generically globally rigid in R^3. Our proof is based on a new sufficient condition for the so called vertex splitting operation to preserve generic global rigidity in R^d.
SageMaths 8.1 code for identifying irreducible (2,2)-tight torus graphs. Complete description of ... more SageMaths 8.1 code for identifying irreducible (2,2)-tight torus graphs. Complete description of all 116 irreducibles
We investigate properties of sparse and tight surface graphs. In particular we derive topological... more We investigate properties of sparse and tight surface graphs. In particular we derive topological inductive constructions for \( (2,2) \)-tight surface graphs in the case of the sphere, the plane, the twice punctured sphere and the torus. In the case of the torus we identify all 116 irreducible base graphs and provide a geometric application to configurations of circular arcs in the spirit of the Koebe-Andreev-Thurston circle packing theorem.
2018 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining (ASONAM), 2018
1 On the WWW, a huge body of data is linked and interlinked in complex ways. The way that informa... more 1 On the WWW, a huge body of data is linked and interlinked in complex ways. The way that information spreads through a network can be examined by creating different graph models and comparing their structural properties. This work examines clustering in network models, which affects how information is spread in a network. In particular, we compare clustering coefficient and transitivity measures in random geometric graphs with those of the Erdös Rényi model of random graphs. Clustering coefficients measure the tendency with which nodes are clustered in a graph. The most striking result is that the transitivity and network average clustering coefficients differ within the geometric graphs. This was unexpected, as these measures coincide in the Erdös Rényi model. This raises potential for using these models in network analysis and in game theory applications.
The influence of our peers is a powerful reinforcement for our social behaviour, evidenced in vot... more The influence of our peers is a powerful reinforcement for our social behaviour, evidenced in voter behaviour and trend adoption. Bootstrap percolation is a simple method for modelling this process. In this work we look at bootstrap percolation on hyperbolic random geometric graphs, which have been used to model the Internet graph, and introduce a form of bootstrap percolation with recovery, showing that random targeting of nodes for recovery will delay adoption, but this effect is enhanced when nodes of high degree are selectively targeted.
arXiv: Combinatorics, 2020
We characterise the quotient surface graphs arising from symmetric contact systems of line segmen... more We characterise the quotient surface graphs arising from symmetric contact systems of line segments in the plane and also from symmetric pointed pseudotriangulations in the case where the group of symmetries is generated by a translation or a rotation of finite order. These results generalise well known results of Thomassen, in the case of line segments, and of Streinu and Haas et al., in the case of pseudotriangulations. Our main tool is a new inductive characterisation of the appropriate classes of surface graphs. We also discuss some consequences of our results in the area of geometric rigidity theory.
arXiv: Combinatorics, 2015
A simple graph is $3$-rigid if its generic bar-joint frameworks in $R^3$ are infinitesimally rigi... more A simple graph is $3$-rigid if its generic bar-joint frameworks in $R^3$ are infinitesimally rigid. Necessary and sufficient conditions are obtained for the minimal $3$-rigidity of a simple graph which is obtained from the $1$-skeleton of a triangulated torus by the deletion of edges interior to a triangulated disc.
We consider the problem of finding an inductive construction, based on vertex splitting, of trian... more We consider the problem of finding an inductive construction, based on vertex splitting, of triangulated spheres with a fixed number of additional edges (braces). We show that for any positive integer b there is such an inductive construction of triangulations with b braces, having finitely many base graphs. In particular we establish a bound for the maximum size of a base graph with b braces that is linear in b. In the case that b = 1 or 2 we determine the list of base graphs explicitly. Using these results we show that doubly braced triangulations are (generically) minimally rigid in two distinct geometric contexts arising from a hypercylinder in R and a class of mixed norms on R.
We consider the problem of finding an inductive construction, based on vertex splitting, of trian... more We consider the problem of finding an inductive construction, based on vertex splitting, of triangulated spheres with a fixed number of additional edges (braces). We show that for any positive integer b there is such an inductive construction of triangulations with b braces, having finitely many base graphs. In particular we establish a bound for the maximum size of a base graph with b braces that is linear in b. In the case that b = 1 or 2 we determine the list of base graphs explicitly. Using these results we show that doubly braced triangulations are (generically) minimally rigid in two distinct geometric contexts arising from a hypercylinder in R and a class of mixed norms on R.
International Mathematics Research Notices, 2018
We consider the problem of characterising the generic rigidity of bar-joint frameworks in $\mathb... more We consider the problem of characterising the generic rigidity of bar-joint frameworks in $\mathbb{R}^d$ in which each vertex is constrained to lie in a given affine subspace. The special case when $d=2$ was previously solved by Streinu and Theran [14] in 2010. We will extend their characterisation to the case when $d\geq 3$ and each vertex is constrained to lie in an affine subspace of dimension $t$, when $t=1,2$ and also when $t\geq 3$ and $d\geq t(t-1)$. We then point out that results on body–bar frameworks obtained by Katoh and Tanigawa [8] in 2013 can be used to characterise when a graph has a rigid realisation as a $d$-dimensional body–bar framework with a given set of linear constraints.
Linear Algebra and its Applications, 2018
Linear Algebra and its Applications, 2018
Journal of Combinatorial Theory, Series B, 2017
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Papers by James Cruickshank