On 4-Valent Symmetric Graphs

AbstractLet G act transitively on incident vertex, edge pairs of the connected 4-valent graph Γ. If a normal subgroup N does not give rise to a natural 4-valent quotient ΓN with G/N acting transitively on incident vertex, edge pairs, then either (a) N has just one or two orbits on vertices, or (b) N has r ⩾ 3 orbits on vertices and the natural quotient ΓN is a circuit Cr (Theorem 1.1). We give a complete classification of the graphs arising in (a) when the normal subgroup N is elementary abelian (Theorems 1.2 and 1.3). Case (b), which depends to some extent on case (a), is more technical and is studied in a subsequent paper

Symmetric Vertex Models on Planar Random Graphs

We solve a 4-(bond)-vertex model on an ensemble of 3-regular Phi3 planar random graphs, which has the effect of coupling the vertex model to 2D quantum gravity. The method of solution, by mapping onto an Ising model in field, is inspired by the solution by Wu et.al. of the regular lattice equivalent -- a symmetric 8-vertex model on the honeycomb lattice, and also applies to higher valency bond vertex models on random graphs when the vertex weights depend only on bond numbers and not cyclic ordering (the so-called symmetric vertex models). The relations between the vertex weights and Ising model parameters in the 4-vertex model on Phi3 graphs turn out to be identical to those of the honeycomb lattice model, as is the form of the equation of the Ising critical locus for the vertex weights. A symmetry of the partition function under transformations of the vertex weights, which is fundamental to the solution in both cases, can be understood in the random graph case as a change of integration variable in the matrix integral used to define the model. Finally, we note that vertex models, such as that discussed in this paper, may have a role to play in the discretisation of Lorentzian metric quantum gravity in two dimensions.Comment: Tidied up version accepted for publication in PL

Symmetric isostatic frameworks with ℓ1\ell^1 or ℓ∞\ell^\infty distance constraints

Combinatorial characterisations of minimal rigidity are obtained for symmetric 2-dimensional bar-joint frameworks with either $\ell^1$ or $\ell^\infty$ distance constraints. The characterisations are expressed in terms of symmetric tree packings and the number of edges fixed by the symmetry operations. The proof uses new Henneberg-type inductive construction schemes.Comment: 20 pages. Main theorem extended. Construction schemes refined. New titl