Abstract
Given a vertex v of a graph G the second order degree of v denoted as d 2(v) is defined as the number of vertices at distance 2 from v. In this paper we address the following question: What are the sufficient conditions for a graph to have a vertex v such that d 2(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 K −4 -free graph or every maximal planar graph is shown to have a vertex v such that d 2(v) ≥ d(v). Other sufficient conditions on graphs for guaranteeing this property are also proved.
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Supported by the Ministry of Education and Science, Spain, the European Regional Development Fund (ERDF) under project MTM2008-06620-C03-02, the Catalan Government under project 2009 SGR 1298, CONACyT-México under project 57371, and PAPIIT-UNAM IN104609-3
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Araujo-Pardo, G., Balbuena, C., Olsen, M. et al. On second order degree of graphs. Acta. Math. Sin.-English Ser. 28, 171–182 (2012). https://doi.org/10.1007/s10114-012-9343-8
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DOI: https://doi.org/10.1007/s10114-012-9343-8