  The degree matrix of a graph is the diagonal matrix with diagonal entries
equal to the degrees of the vertices of $X$. If $X_1$ and $X_2$ are graphs with
respective adjacency matrices $A_1$ and $A_2$ and degree matrices $D_1$ and
$D_2$, we say that $X_1$ and $X_2$ are degree similar if there is an invertible
real matrix $M$ such that $M^{-1}A_1M=A_2$ and $M^{-1}D_1M=D_2$. If graphs
$X_1$ and $X_2$ are degree similar, then their adjacency matrices, Laplacian
matrices, unsigned Laplacian matrices and normalized Laplacian matrices are
similar. We first show that the converse is not true. Then, we provide a number
of constructions of degree-similar graphs. Finally, we show that the matrices
$A_1-\mu D_1$ and $A_2-\mu D_2$ are similar over the field of rational
functions $\mathbb{Q}(\mu)$ if and only if the Smith normal forms of the
matrices $tI-(A_1-\mu D_1)$ and $tI-(A_2-\mu D_2)$ are equal.
