A note on Second Degrees in Graphs - Naji, Ahmed Mohammed (Indonesian Journal of Combinatorics, 2023)
The second degree of a node x in a graph Γ=(V,E), denoted by deg2(x), is the number of nodes at distance two from x in a graph Γ. 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 "deg2(x) ≤  (∑(y ∈ N(x)) deg(y)) - deg(x), for every x ∈ V(Γ)", 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.