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José Cáceres 《Discrete Mathematics》2006,306(2):188-198
Let G be a finite simple connected graph. A vertex v is a boundary vertex of G if there exists a vertex u such that no neighbor of v is further away from u than v. We obtain a number of properties involving different types of boundary vertices: peripheral, contour and eccentric vertices. Before showing that one of the main results in [G. Chartrand, D. Erwin, G.L. Johns, P. Zhang, Boundary vertices in graphs, Discrete Math. 263 (2003) 25-34] does not hold for one of the cases, we establish a realization theorem that not only corrects the mentioned wrong statement but also improves it.Given S⊆V(G), its geodetic closure I[S] is the set of all vertices lying on some shortest path joining two vertices of S. We prove that the boundary vertex set ∂(G) of any graph G is geodetic, that is, I[∂(G)]=V(G). A vertex v belongs to the contour Ct(G) of G if no neighbor of v has an eccentricity greater than v. We present some sufficient conditions to guarantee the geodeticity of either the contour Ct(G) or its geodetic closure I[Ct(G)]. 相似文献
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The eccentricity of a vertex v in a graph is the maximum of the distances from v to all other vertices. The diameter of a graph is the maximum of the eccentricities of its vertices. Fix the parameters n, d, c. Over all graphs with order n and diameter d, we determine the maximum (within 1) and the minimum of the number of vertices with eccentricity c.
Revised: May 7, 1999 相似文献
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Dynamics of multiphase flow under high voltage has attracted extensive research interests due to its wide industrial applications. In this paper, numerical solution of electro-hydrodynamic behavior and interface instability of double emulsion droplet is presented. Level set method and leaky dielectric model coupled with Navier-Stokes equation are used to solve the electro-hydrodynamic problem. The method is validated against the theoretical analysis and the simulation results of the other researchers. Double emulsion droplet with inner droplet (core) and outer droplet (shell) phases immersed in continuous phase is subjected to high electric field. Shell/continuous and core/shell interfaces of the droplet undergo prolate-oblate or oblate-prolate deformation depending on the extent of the penetration of electric potential and sense of charge distribution at the interfaces. The deformation of the shell deviates from theory at larger volume fraction of core for oblate-prolate case whereas it follows theory for prolate-oblate case. The interfaces showing oblate-prolate deformation split at the poles whereas, for prolate-oblate, they split away along the equator. The re-union of the interfaces under high electric field results with production of daughter droplet at the core. The large decrease in critical electric field for oblate-prolate case shows their less interface stability at larger volume fraction of core. When the core is eccentric, the electric field drives it towards the shell center or to the shell/continuous interface depending on electrical parameters. The study is beneficial in understanding the electro-hydrodynamic behavior of emulsion droplets and efficient design of related industrial processes. 相似文献
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The eccentric distance sum (EDS) is a novel topological index that offers a vast potential for structure activity/property relationships. For a connected graph G, the eccentric distance sum is defined as ξd(G)=∑v∈V(G)ecG(v)DG(v), where ecG(v) is the eccentricity of a vertex v in G and DG(v) is the sum of distances of all vertices in G from v. More recently, Yu et al. [G. Yu, L. Feng, A. Ili?, On the eccentric distance sum of trees and unicyclic graphs, J. Math. Anal. Appl. 375 (2011) 99-107] proved that for an n-vertex tree T, ξd(T)?4n2−9n+5, with equality holding if and only if T is the n-vertex star Sn, and for an n-vertex unicyclic graph G, ξd(G)?4n2−9n+1, with equality holding if and only if G is the graph obtained by adding an edge between two pendent vertices of n-vertex star. In this note, we give a short and unified proof of the above two results. 相似文献