1,164 research outputs found
Determination of the star valency of a graph
AbstractThe star valency of a graph G is the minimum, over all star decompositions π, of the maximum number of elements in π incident with a vertex. The maximum average degree of G, denoted by dmax-ave(G), is the maximum average degree of all subgraphs of G. In this paper, we prove that the star valency of G is either ⌈dmax-ave(G)/2⌉ or ⌈dmax-ave(G)/2⌉+1, and provide a polynomial time algorithm for determining the star valency of a graph
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
Characterising vertex-star transitive and edge-star transitive graphs
Recent work of Lazarovich provides necessary and sufficient conditions on a graph L for there to exist a unique simply-connected (k, L)-complex. The two conditions are symmetry properties of the graph, namely vertex-star transitivity and edge-star transitivity. In this paper we investigate vertex- and edge-star transitive graphs by studying the structure of the vertex and edge stabilisers of such graphs. We also provide new examples of graphs that are both vertex-star transitive and edge-star transitive
Roots, symmetries and conjugacy of pseudo-Anosov mapping classes
An algorithm is proposed that solves two decision problems for pseudo-Anosov
elements in the mapping class group of a surface with at least one marked fixed
point. The first problem is the root problem: decide if the element is a power
and in this case compute the roots. The second problem is the symmetry problem:
decide if the element commutes with a finite order element and in this case
compute this element. The structure theorem on which this algorithm is based
provides also a new solution to the conjugacy problem
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