Total eccentricity index of trees with fixed pendent vertices and trees with fixed diameter

Author(s):  
Rashid Farooq ◽  
Shehnaz Akhter ◽  
Juan Rada
2019 ◽  
Vol 255 ◽  
pp. 339-344
Author(s):  
Hailiang Zhang ◽  
Guangting Chen ◽  
Guanglong Yu
Keyword(s):  

Author(s):  
Fazal Hayat

The connective eccentricity index (CEI for short) of a graph [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the degree of [Formula: see text] and [Formula: see text] is the eccentricity of [Formula: see text] in [Formula: see text]. In this paper, we characterize the unique graphs with maximum CEI from three classes of graphs: the [Formula: see text]-vertex graphs with fixed connectivity and diameter, the [Formula: see text]-vertex graphs with fixed connectivity and independence number, and the [Formula: see text]-vertex graphs with fixed connectivity and minimum degree.


2010 ◽  
Vol 4 (1) ◽  
pp. 181-196 ◽  
Author(s):  
Nair Abreu ◽  
Oscar Rojo ◽  
Claudia Justel

A caterpillar is a tree in which the removal of all pendent vertices make it a path. In this paper, we consider two classes of caterpillars. We present an ordering of caterpillars by algebraic connectivity in one of them and find one that maximizes the algebraic connectivity in the other class.


2021 ◽  
Vol 42 (Supplement_1) ◽  
Author(s):  
B Goebel ◽  
C Salomon ◽  
M Abdulrahman ◽  
S Richter ◽  
M El Garhy ◽  
...  

Abstract Background The concept of percutaneous tricuspid valve edge-to-edge repair (pTVR) is based on the connection of leaflets in the area of insufficiency using a coaptation device. By closing the coaptation device a considerable tractive force is applied on the leaflets, which might have an effect on the valve ring. Aim of the study was to examine the impact of device implantation on tricuspid ring dimensions. Methods During pTVR 3D zoom loops of tricuspid valve were acquired before and after clip placement using transoesophageal echocardiography. Measurements of TV ring dimensions included the following parameters: ring area (TV area), maximal diameter, minimal diameter, eccentricity index (Figure 1). Tenting area was derived from a four-chamber view of the valve. In addition, regurgitation severity was graded from 1+ to 5+ by measuring vena contracta area (VCA3D) in 3D full volume colour Doppler loop using multiplanar reconstruction. Right atrial (RA) and ventricular volumes (RVVd3D, RVVs3D) and function (RVEF3D) were assessed in a 3D full volume loop. Results The study population comprised 97 patients (age 78±6 years, 47 male), who underwent pTVR at our hospital. As expected cavity dimension correlated with TV area size (for RVVd3D r=0.51, p<0.001 and for RA volume r=0.71, p<0.001). The mean TV ring area was significantly reduced (ring area 8.53±2.23 cm2/m2BSA vs. 7.55±2.18 cm2/m2BSA, p<0.001) and the ring shape became more oval (Eccentricity index 1.2±0.15 vs. 1.29±0.17, p<0.001) after pTVR. The reduction of ring area (12±7%, range 0.7–28%) showed an only modest correlation to the number of implanted coaptation devices (r=0.30, p<0.001) and percentage reduction of VCA3D (r=0.36, p<0.001). In the patient group with a ring area change ≥12% a reduction to TR grade ≤2+ by pTVR was achieved in 83% of cases, whereas only 62% of patients reached moderate TR when area change was below 12%. Conclusion pTVR using coaptation devices reduces the ring area. This effect is related to the number of devices implanted. FUNDunding Acknowledgement Type of funding sources: None.


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