  A Deza graph $G$ with parameters $(n,k,b,a)$ is a $k$-regular graph with $n$
vertices such that any two distinct vertices have $b$ or $a$ common neighbours.
The children $G_A$ and $G_B$ of a Deza graph $G$ are defined on the vertex set
of $G$ such that every two distinct vertices are adjacent in $G_A$ or $G_B$ if
and only if they have $a$ or $b$ common neighbours, respectively. A strongly
Deza graph is a Deza graph with strongly regular children. In this paper we
give a spectral characterisation of strongly Deza graphs, show relationships
between eigenvalues, and study strongly Deza graphs which are distance-regular.
