On the maximum weight of a planar graph of given order and size

The weight of an edge uvuv of a graph is defined to be the sum of degrees of the vertices uu and vv. The weight of a non-empty graph GG is the minimum of the weights of edges of GG. The paper is concerned with the maximum weight of a planar graph having nn vertices and mm edges. It is shown that if m≥2n+1m≥2n+1, then the maximum weight is at most ⌊9m−12nm−2n⌋. Moreover, there are infinitely many pairs (n,m)(n,m) such that the maximum weight is at least ⌊9m−12nm−2n⌋−1.