On a conjecture concerning integral real roots of certain cubic polynomials

In this paper, we determine all the rational pairs (x,n) such thatfn(x)=3x3+(7−10n)x2+2(6n2−11n+8)x−(4n3−6n2−10n+24)=0. It follows that for each positive integer n⩾5, there is no integer solution x for the polynomial. This confirms a conjecture of Li et al. concerning the uniqueness of a class of optimal graphs in the study of an extremal graph theory problem.