A survey of the use of graph theoretical techniques in Biology is presented. In particular, recent work on identifying and modelling the structure of bio-molecular networks is discussed, as well as the application of centrality measures to interaction networks and research on the hierarchical structure of such networks and network motifs. Work on the link between structural network properties and dynamics is also described, with emphasis on synchronisation and disease propagation.

We present new necessary and sufficient conditions for the existence of fixed points in a finite system of coupled phase oscillators on a complete graph. We use these conditions to derive bounds on the critical coupling.

We extend recent results [50] on the existence of global phase-locked states (GPLS) in the Kuramoto model on a complete graph to the case of a complete bipartite graph. In particular, we prove that, for the Kuramoto model on a complete bipartite graph, the value of the critical coupling coefficient can be determined by solving a system of two nonlinear equations that do not depend on the coupling coefficient. We show that the said system of equations can be solved using an efficient algorithm described in the paper.

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We extend two fundamental properties of positive linear time-invariant (LTI) systems to homogeneous cooperative systems. Specifically, we demonstrate that such systems are -stable, meaning that global asymptotic stability is preserved under diagonal scaling. We also show that a delayed homogeneous cooperative system is globally asymptotically stable (GAS) for any non-negative delay if and only if the system is GAS for zero delay.