Sums of valences in bigraphs. II

In the first paper of this title it was shown that the sum of the valences at three distinct vertices of a bigraph on 2 n points obeyed v p + v q + v r ⩽ 3n − 1, if n = 2m, 3n, if n = 2m − 1 A class of bigraphs (modified central star and central star) was defined which demonstrated that these bounds were the best possible by achieving equality in each case. In this note, it is shown that for n odd the central star bigraph and for n even the modified central star and three simple variations on the central star are the only bigraphs having sums of valences which achieve the bound and that for all other bigraphs strict inequality prevails. It is also noted that this result implies that the sum of valences at m distinct vertices of a bigraph is strictly less than mn for all m ≤ 4.