We describe all the trees with the property that the corresponding edge ideal of the square of the tree has a linear resolution. As a consequence, we give a complete characterization of those trees $T$ for which the square is co-chordal, that is the complement of the square, $(T^2)^c$, is a chordal graph. For particular classes of trees such as paths and double brooms, we determine the Krull dimension and the projective dimension.
"Edge ideals of squares of trees." Osaka J. Math. 59 (2) 369 - 386, April 2022.