| 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. |
