  Let $X$ be a tree with $n$ vertices and $L(X)$ be its line graph. In this
work, we completely characterize the trees for which the algebraic connectivity
of $L(X)\times K_m$ is equal to $m-1$, where $\times$ denotes the Kronecker
product. We provide a few necessary and sufficient conditions for $L(X)\times
K_m$ to be Laplacian integral. The algebraic connectivity of $L(X)\times K_m$,
where $X$ is a tree of diameter $4$ and $k$-book graph is discussed.
