  In this paper, we give a further study on $B$-tensors and introduce doubly
$B$-tensors that contain $B$-tensors. We show that they have similar
properties, including their decompositions and strong relationship with
strictly (doubly) diagonally dominated tensors. As an application, the
properties of $B$-tensors are used to localize real eigenvalues of some
tensors, which would be very useful in verifying the positive semi-definiteness
of a tensor.
