  Let $M\circ N$ denote the Schur product of two matrices $M$ and $N$. A graph
$X$ with adjacency matrix $A$ is walk regular if $A^k\circ I$ is a constant
times $I$ for each $k\ge0$, and $X$ is 1-walk-regular if it is walk regular and
$A^k\circ A$ is a constant times $A$ for each $k\ge0$. Assume $X$ is 1-walk
regular. Here we show that by deleting an edge in $X$, or deleting edges of a
graph inside a clique of $X$, we obtain families of graphs that are not
necessarily isomorphic, but are cospectral with respect to four types of
matrices: the adjacency matrix, Laplacian matrix, unsigned Laplacian matrix,
and normalized Laplacian matrix. Furthermore, we show that removing edges of
Laplacian cospectral graphs in cliques of a 1-walk regular graph results in
Laplacian cospectral graphs; removing edges of unsigned Laplacian cospectral
graphs whose complements are also cospectral with respect to the unsigned
Laplacian in cliques of a 1-walk regular graph results in unsigned Laplacian
cospectral graphs.
