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

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