Joanna B. Fawcett

Stochastic cycle selection in active flow networks

Significance Nature often uses interlinked networks to transport matter or information. In many cases, the physical or virtual flows between network nodes are actively driven, consuming energy to achieve transport along different links. However, there are currently very few elementary principles known to predict the behavior of this broad class of nonequilibrium systems. Our work develops a generic foundational understanding of mass-conserving active flow networks. By merging previously disparate concepts from mathematical graph theory and physicochemical reaction rate theory, we relate the self-organizing behavior of actively driven flows to the fundamental topological symmetries of the underlying network, resulting in a class of predictive models with applicability across many biological scales. Active biological flow networks pervade nature and span a wide range of scales, from arterial blood vessels and bronchial mucus transport in humans to bacterial flow through porous media or plasmodial shuttle streaming in slime molds. Despite their ubiquity, little is known about the self-organization principles that govern flow statistics in such nonequilibrium networks. Here we connect concepts from lattice field theory, graph theory, and transition rate theory to understand how topology controls dynamics in a generic model for actively driven flow on a network. Our combined theoretical and numerical analysis identifies symmetry-based rules that make it possible to classify and predict the selection statistics of complex flow cycles from the network topology. The conceptual framework developed here is applicable to a broad class of biological and nonbiological far-from-equilibrium networks, including actively controlled information flows, and establishes a correspondence between active flow networks and generalized ice-type models.

LOCALLY TRIANGULAR GRAPHS AND RECTAGRAPHS WITH SYMMETRY

Abstract Locally triangular graphs are known to be halved graphs of bipartite rectagraphs, which are connected triangle-free graphs in which every 2-arc lies in a unique quadrangle. A graph Γ is locally rank 3 if there exists G ⩽ Aut ( Γ ) such that for each vertex u , the permutation group induced by the vertex stabiliser G u on the neighbourhood Γ ( u ) is transitive of rank 3. One natural place to seek locally rank 3 graphs is among the locally triangular graphs, where every induced neighbourhood graph is isomorphic to a triangular graph T n . This is because the graph T n , which has vertex set the 2-subsets of { 1 , … , n } and edge set the pairs of 2-subsets intersecting at one point, admits a rank 3 group of automorphisms. In this paper, we classify a certain family of rectagraphs for which the permutation group induced by Aut ( Γ ) u on Γ ( u ) is 4-homogeneous for some vertex u . We then use this result to classify the connected locally triangular graphs that are also locally rank 3.

Primitive permutation groups with a suborbit of length 5 and vertex-primitive graphs of valency 5

Abstract We classify finite primitive permutation groups having a suborbit of length 5. As a corollary, we obtain a classification of finite vertex-primitive graphs of valency 5. In the process, we also classify finite almost simple groups that have a maximal subgroup isomorphic to Alt ( 5 ) or Sym ( 5 ) .

Locally triangular graphs and normal quotients of the n-cube

For an integer

On

n\ge 2

-connected-homogeneous graphs

n≥2, the triangular graph has vertex set the 2-subsets of