The Maximal Subgroups and the Complexity of the Flow Semigroup of Finite (Di)graphs

Preprint of an article first published online in International Journal of Algebra and Computation, September 2017, doi: https://doi.org/10.1142/S0218196717500412. © 2017 Copyright World Scientific Publishing Company. http://www.worldscientific.com/worldscinet/ijac. Accepted Manuscript version is under embargo. Embargo end date: 26 September 2018.The flow semigroup, introduced by John Rhodes, is an invariant for digraphs and a complete invariant for graphs. We refine and prove Rhodes's conjecture on the structure of the maximal groups in the flow semigroup for finite, antisymmetric, strongly connected graphs. Building on this result, we investigate and fully describe the structure and actions of the maximal subgroups of the flow semigroup acting on all but k points for all finite digraphs and graphs for all k >=1. A linear algorithm is presented to determine these so-called 'defect k groups' for any finite (di)graph. Finally, we prove that the complexity of the flow semigroup of a 2-vertex connected (and strongly connected di)graph with n vertices is n- 2, completely confirming Rhodes's conjecture for such (di)graphs.Peer reviewe

Intravenous apomorphine therapy in Parkinson's disease - Clinical and pharmacokinetic observations

Six patients with Parkinson's disease and refractory motor fluctuations, with severe subcutaneous (s,c,) nodule formation as a result of long-term s,c, apomorphine infusions, were switched to intravenous (i,v,) therapy via a long-term in-dwelling venous catheter, Five patients were followed-up for a mean of 7 months (range 0,5-18 months). All patients had plasma apomorphine concentrations measured at baseline during s,c, infusions and three had follow-up measurements when stabilized on i,v, therapy, to test the hypothesis that motor fluctuations in these patients are largely due to impaired absorption of apomorphine, The mean i,v, rate of 9.0 mg/h (range 5-14 mg) and 24-h dose of 256.7 mg (range 90-456 mg) of apomorphine were not significantly reduced compared with the s,c, route (9.24 mg/h and 243.4 mg), However, additional oral anti-parkinsonian medication was reduced by a mean of 59%, and 'off' time was virtually eliminated (mean reduction from 5.4 to 0.5 h per day, P < 0,05), There was also a significant reduction in dyskinesias and markedly improved quality of life. Pharmacokinetic analysis demonstrated more reliable and smoother delivery of apomorphine via the i,v, route, although 'off' periods were not always explained by low plasma apomorphine concentrations. Complication rates were high and included three unforeseen hazardous intravascular thrombotic complications, secondary to apomorphine crystal accumulation, necessitating cardiothoracic surgery. We conclude that i,v, apomorphine therapy holds promise as a more effective way of controlling motor fluctuations than the s,c, route. However, further preclinical research is required before i,v, Britaject apomorphine can be recommended for routine clinical practice. Even when stable plasma apomorphine concentrations were achieved, motor fluctuations could not be totally eradicated, suggesting that postsynaptic receptor changes may also play a role in the refractory 'off' periods in these patients