### A note on Second Degrees in Graphs

#### Abstract

The second degree of a node *x* in a graph *Γ*=(*V*,*E*), denoted by deg_{2}(*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 "deg_{2}(*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.

#### Keywords

#### Full Text:

PDFDOI: http://dx.doi.org/10.19184/ijc.2023.7.2.3

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