Graph functions maximized on a path

Given a connected graph G\of order $n$ and a nonnegative symmetric matrix $A=\left[a_{i,j}\right]$ of order $n,$ define the function $F_{A}\left( G\right) $ as% $$F_A\left(G\right)=\sum _{1\le i<j\le n}{d}_G(i,j)a_{i,j},$$ where $d_G(i,j)$ denotes the distance between the vertices $i$ and $j$ in $G\mathrm{.}$ In this note it is shown that $F_{A}\left( G\right) \leq F_{A}\left( P\right) ,$forsomepathoforder$n.$Moreover,ifeachrowof$A$ has at most one zero off-diagonal entry, then $F_{A}\left( G\right) <F_{A}\left( P\right) ,$forsomepathoforder$n,$unless$G$ itself is a path. In particular, this result implies two conjectures of Aouchiche and Hansen:

• the spectral radius of the distance Laplacian of a connected graph $G$ of order $n$ is maximal if and only if $G$ is a path;
• the spectral radius of the distance signless Laplacian of a connected graph $G$ of order $n$ is maximal if and only if $G$ is a path.