Given a vertex v of a graph G the second order degree of v denoted as d2（v） is defined as the number of vertices at distance 2 from v. In this paper we address the following question： What axe the sufficient condit...Given a vertex v of a graph G the second order degree of v denoted as d2（v） is defined as the number of vertices at distance 2 from v. In this paper we address the following question： What axe the sufficient conditions for a graph to have a vertex v such that d2（v） ≥ d（v）, where d（v） denotes the degree of v？ Among other results, every graph of minimum degree exactly 2, except four graphs, is shown to have a vertex of second order degree as large as its own degree. Moreover, every K4^--free graph or every maximal planar graph is shown to have a vertex v such that d2（v） ≥ d（v）. Other sufficient conditions on graphs for guaranteeing this property axe also proved.展开更多
基金Supported by the Ministry of Education and Science,Spainthe European Regional Development Fund (ERDF)under project MTM2008-06620-C03-02+2 种基金the Catalan Government under project 2009 SGR 1298CONACyTMxico under project 57371PAPIIT-UNAM IN104609-3
文摘Given a vertex v of a graph G the second order degree of v denoted as d2（v） is defined as the number of vertices at distance 2 from v. In this paper we address the following question： What axe the sufficient conditions for a graph to have a vertex v such that d2（v） ≥ d（v）, where d（v） denotes the degree of v？ Among other results, every graph of minimum degree exactly 2, except four graphs, is shown to have a vertex of second order degree as large as its own degree. Moreover, every K4^--free graph or every maximal planar graph is shown to have a vertex v such that d2（v） ≥ d（v）. Other sufficient conditions on graphs for guaranteeing this property axe also proved.
