A note on Second Degrees in Graphs

Abstract

<p>The second degree of a node <em>x</em> in a graph <em>Γ</em>=(<em>V</em>,<em>E</em>), denoted by deg<sub>2</sub>(<em>x</em>), is the number of nodes at distance two from <em>x</em> in a graph <em>Γ</em>. In the present article, we are interested in examination of the second degrees properties in a graph. The old bounds and the general formulas of the second degree of some graph operations are collected. We provide an improvement on the useful result "deg<sub>2</sub>(<em>x</em>) ≤  (∑<sub>(<em>y</em> ∈ <em>N</em>(<em>x</em>))</sub> deg(y)) - deg(<em>x</em>), for every <em>x </em>∈ <em>V</em>(<em>Γ</em>)", by adding a term of the triangles number in a graph, in order to the equality holds for each quadrangle-free graph. Further, upper and lower bounds for the maximum and minimum second degrees are established. Finally the second degree-sum formula are derived. In addition, bounds on second degree-sum are also established.</p>