We consider the problem of finding small distance-preserving subgraphs of undirected, unweighted interval graphs with k terminal vertices. We prove the following results. 1. Finding an optimal distance-preserving subgraph is NP-hard for general graphs. 2. Every interval graph admits a subgraph with O(k) branching vertices that approximates pairwise terminal distances up to an additive term of +...

Immunization for complex network based on the effective degree of vertex Ke Hu and Yi Tang Department of Physics and Institute of Modern Physics, Xiangtan University, Xiangtan 411105, Hunan, China The basic idea of many effective immunization strategies is first to rank the importance of vertices according to the degrees of vertices and then remove the vertices from highest importance to lowest...

For a planar graph on n vertices we determine the maximum values for the following: 1) the sum of the m largest vertex degrees. 2) the number of vertices of degree at least k. 3) the sum of the degrees of vertices with degree at least k.

Astigmatism’s defect has not yet been formulated in Physics and medical physics books theoretically. In this paper, we are trying to formulate the astigmatism’s defect. The results of formulation show that “the degree of astigmatism is negative reverse the distance that vision’s condition of astigmatism’s chart lines change conditions in dimness angle from dimness condition at infinite distance...

Astigmatism’s defect has not yet been formulated in Physics and medical physics books theoretically. In this paper, we are trying to formulate the astigmatism’s defect. The results of formulation show that “the degree of astigmatism is negative reverse the distance that vision’s condition of astigmatism’s chart lines change conditions in dimness angle from dimness condition at infinite distance...

We compute the distance-dependent two-point function of vertex-bicolored planar maps, i.e., maps whose vertices are colored in black and white so that no adjacent vertices have the same color. By distance-dependent two-point function, we mean the generating function of these maps with both a marked oriented edge and a marked vertex which are at a prescribed distance from each other. As customar...

Let H be a hypergraph with n vertices. Suppose that d1, d2, . . . , dn are degrees of the vertices of H. The t-th graph entropy based on degrees of H is defined as