  We present a geometrical construction of families of finite isospectral
graphs labelled by different partitions of a natural number $r$ of given length
$s$ (the number of summands). Isospectrality here refers to the discrete
magnetic Laplacian with normalised weights (including standard weights). The
construction begins with an arbitrary finite graph $G$ with normalised weight
and magnetic potential as a building block from which we construct, in a first
step, a family of so-called frame graphs $(F_a)_{a \in \mathbb{N}}$. A frame
graph $F_a$ is constructed contracting $a$ copies of $G$ along a subset of
vertices $V_0$. In a second step, for any partition $A=(a_1,\dots,a_s)$ of
length $s$ of a natural number $r$ (i.e., $r=a_1+\dots+a_s$) we construct a new
graph $F_A$ contracting now the frames $F_{a_1},\dots,F_{a_s}$ selected by $A$
along a proper subset of vertices $V_1\subset V_0$. All the graphs obtained by
different $s$-partitions of $r\geq 4$ (for any choice of $V_0$ and $V_1$) are
isospectral and non-isomorphic.
  In particular, we obtain increasing finite families of graphs which are
isospectral for given $r$ and $s$ for different types of magnetic Laplacians
including the standard Laplacian, the signless standard Laplacian, certain
kinds of signed Laplacians and, also, for the (unbounded) Kirchhoff Laplacian
of the underlying equilateral metric graph. The spectrum of the isospectral
graphs is determined by the spectrum of the Laplacian of the building block $G$
and the spectrum for the Laplacian with Dirichlet conditions on the set of
vertices $V_0$ and $V_1$ with multiplicities determined by the numbers $r$ and
$s$ of the partition.
