Counting irregular multigraphs

AbstractGagliardi et al. (1996, unpublished manuscript) defined an irregular multigraph to be a loopless multigraph with degree sequence n, n − 1,…, 1, and they posed the problem of determining the number of different irregular multigraphs fn on n vertices. In Gagliardi et al. (1996) they showed that if n ≡ 0 or 3 (mod 4) then fn > n − 1. In this note our aim is to show that there are constants 1 < c1 < c2 and n0 > 0 such that if n ⩾ n0 and n ≡ 0 or 3 (mod 4) then (c1)n2 < fn < (c2)n2. Indeed, we show that c1 = 1.19 and c2 = 1.65 can be chosen

Contact force sensing in ablation of ventricular arrhythmias using a 56-hole open-irrigation catheter: a propensity-matched analysis.

PURPOSE: The effect of adding contact force (CF) sensing to 56-hole tip irrigation in ventricular arrhythmia (VA) ablation has not been previously studied. We aimed to compare outcomes with and without CF sensing in VA ablation using a 56-hole radiofrequency (RF) catheter. METHODS: A total of 164 patients who underwent first-time VA ablation using Thermocool SmartTouch Surround Flow (TC-STSF) catheter (Biosense-Webster, Diamond Bar, CA, USA) were propensity-matched in a 1:1 fashion to 164 patients who had first-time ablation using Thermocool Surround Flow (TC-SF) catheter. Patients were matched for age, gender, cardiac aetiology, ejection fraction and approach. Acute success, complications and long-term follow-up were compared. RESULTS: There was no difference between procedures utilising either TC-SF or TC-STSF in acute success (TC-SF: 134/164 (82%), TC-STSF: 141/164 (86%), p = 0.3), complications (TC-SF: 11/164 (6.7%), TC-STSF: 11/164 (6.7%), p = 1.0) or VA-free survival (TC-SF: mean arrhythmia-free survival time = 5.9 years, 95% CI = 5.4-6.4, TC-STSF: mean = 3.2 years, 95% CI = 3-3.5, log-rank p = 0.74). Fluoroscopy time was longer in normal hearts with TC-SF (19 min, IQR: 14-30) than TC-STSF (14 min, IQR: 8-25; p = 0.04). CONCLUSION: Both TC-SF and TC-STSF catheters are safe and effective in treating VAs. The use of CF sensing catheters did not improve safety or acute and long-term outcomes, but reduced fluoroscopy time in normal heart VA