The First Eccentric Zagreb Index of Linear Polycene Parallelogram of Benzenoid

Let G = (V,E) be a graph, where V(G) is a non-empty set of vertices and E(G) is a set of edges, e = uv∈E(G), d(u) is degree of vertex u. Then the first Zagreb polynomial and the first Zagreb index Zg1(G,x) and Zg1(G) of the graph G are defined as Σuv∈E(G)x(du+dv) and Σe=uv∈E(G)(du+dv) respectively. Recently Ghorbani and Hosseinzadeh introduced the first Eccentric Zagreb index as Zg1*=Σuv∈E(G)(ecc(v)+ecc(u)), that ecc(u) is the largest distance between u and any other vertex v of G. In this paper, we compute this new index (the first Eccentric Zagreb index or third Zagreb index) of an infinite family of linear Polycene parallelogram of benzenoid.

On the ECI and CEI of (3, 6)-Fullerenes

The eccentricity of a vertex in a graph is the maximum distance from the vertex to any other vertex. Two structure topological indices: eccentric connectivity index and connective eccentricity index involving eccentricity have a wide range of applications in structure-activity relationships and pharmaceutical drug design etc. In this paper, we investigate the eccentric connectivity index and the connective eccentricity index of a (3, 6)-fullerene. We find a relation between the radius and the number of spokes of a (3, 6)-fullerene. Based on the relation, we give the computing formulas of the eccentric connectivity index and the connective eccentricity index of a (3, 6)-fullerene, respectively.

Rheumatic Aortic Valve Disease with Mitral Stenosis—A Case Report

Congenitally malformed aortic valves are more susceptible to valve injury due to rheumatic fever, mechanical stress of altered flow patterns, atherosclerotic risk factors and degenerative changes. Rheumatic involvement usually occurs in childhood and it is progressive leading to diffuse thickening and fibrosis at leaflet edges and thus differentiated from other patterns of valve damage. Background of this case report revealed the bicuspid nature of the aortic valve due to rheumatic commissural fusion and analysis of echocardiographic parameters in combined lesions of both aortic and mitral valves with severe LV (left ventricular) dysfunction. Left ventricular (LV) and left atrial (LA) dilations predisposing to the formation of smoke (SEC-spontaneous echo contrast) in LV and LA as a consequence of mitral and aortic valve disease are illustrated by 2D echocardiographic imaging in this 41-year-old male.

On the maximal eccentric connectivity indices of graphs

For a connected simple graph G, the eccentricity ec（v） of a vertex v in G is the distance from v to a vertex farthest from v, and d（v） denotes the degree of a vertex v. The eccentric connectivity index of G, denoted by ξC（G）, is defined as ∑vЕV（G） d（v）ec（v）. In this paper, we will determine the graphs with maximal eccentric connectivity index among the connected graphs with n vertices and m edges（n ≤ m ≤ n ＋ 4）, and propose a conjecture on the graphs with maximal eccentric connectivity index and m edges （m ≥ n ＋ 5）. among the connected graphs with n vertices

Comparison and Extremal Results on Three Eccentricity-based Invariants of Graphs

The first and second Zagreb eccentricity indices of graph G are defined as:E1(G)=∑(vi)∈V(G)εG(vi)~2,E2(G)=∑(vivj)∈E(G)εG(vi)εG(vj)whereεG(vi)denotes the eccentricity of vertex vi in G.The eccentric complexity C(ec)(G)of G is the number of different eccentricities of vertices in G.In this paper we present some results on the comparison between E1(G)/n and E2(G)/m for any connected graphs G of order n with m edges,including general graphs and the graphs with given C(ec).Moreover,a Nordhaus-Gaddum type result C(ec)(G)+C(ec)(■)is determined with extremal graphs at which the upper and lower bounds are attained respectively.

On a Novel Eccentricity-based Invariant of a Graph

In this paper, for the purpose of measuring the non-self-centrality extent of non-self- centered graphs, a novel eccerttricity-based invariant, named as non-self-centrality number （NSC num- ber for short）, of a graph G is defined as follows： N（G） =∑vi,vj∈V（G）｜ei-ej｜ where the summation goes over all the unordered pairs of vertices in G and ei is the eccentricity of vertex vi in G, whereas the invariant will be called third Zagreb eccentricity index if the summation only goes over the adja- cent vertex pairs of graph G. In this paper, we determine the lower and upper bounds on N（G） and characterize the corresponding graphs at which the lower and upper bounds are attained. Finally we propose some attractive research topics for this new invariant of graphs.

Comparative study on hemodynamic environments around patient-specific carotid atherosclerotic plaques with different symmetrical features

Atherosclerotic plaque near the carotid sinus is a key risk factor for ischemic stroke.Although disturbed hemodynamic environments around atherosclerotic plaque had been investigated in many studies,the effect of their symmetrical features,especially longitudinal asymmetrical characteristics,had not been comparatively studied.In this study,three-dimensional carotid bifurcation models were established based on CT images of three patients with 50%stenosis and with different symmetrical features of carotid atherosclerotic plaques including concentric plaque,eccentric plaque,or eccentric longitudinal asymmetrical plaque.A healthy subject was chosen as the control.Wall shear stress(WSS)and oscillating shear index in the regions of upstream,downstream,downstream shoulders and regions around the point of maximum stenosis were analyzed.The results indicated that the maximum WSS around the eccentric longitudinal asymmetrical plaque were 8.6 times larger than that of the eccentric plaque and 1.9 times larger than that of the concentric plaque.And the distributions of WSS and OSI around the eccentric longitudinal asymmetric plaque were more disordered than those of the others.Our results highlighted the risk of the eccentric longitudinal asymmetrical plaque on plaque development and rupture and might provide a new index for evaluating the development of atherosclerosis.