A note on Second Degrees in Graphs

Ahmed Mohammed Naji

Abstract

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) ≤  (∑(yN(x)) deg(y)) - deg(x), for every 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

Second degree (of vertex); Distance in graph; Graph operations.

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

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