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% $F_{A}\left( G\right) =\sum_{1\leq i<j\leq n}d_{G}\left( i,j\right) a_{i,j},$ where $d_{G}\left( i,j\right)$ denotes the distance between the vertices $i$ and $j$ in $G.$ In this note it is shown that $F_{A}\left( G\right) \leq F_{A}\left( P\right) \,$for some path of order $n.$ Moreover, if each row of $A$ has at most one zero off-diagonal entry, then $F_{A}\left( G\right) <F_{A}\left( P\right) \,$for some path of order $n,$ 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.

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