  We discuss the construction of $n$-qubit pure states with maximum bipartite
entanglement across all possible choices of $k$ vs $n-k$ bi-partitioning, which
implies that the Von Neumann entropy of every $k$-qubit reduced density matrix
corresponding to this state should be $k \ln 2 $. Such states have been
referred to as $k$-uniform, $k$-MM states. We show that a subset of the 'graph
states' satisfy this condition, hence providing a recipe for constructing
$k$-uniform states. Finding recipes for construction of $k$-uniform states
using graph states is useful since every graph state can be constructed
starting from a product state using only controlled-$Z$ gates. Though, a priori
it is not clear how to construct a graph which corresponds to an arbitrary
$k$-uniform state, but in particular, we show that graphs with no isolated
vertices are $1$-uniform. Graphs organized as a circular linear chain
corresponds to the case of $2$-uniform state, where we show that the minimum
number of qubits required to host such a state is $n=5$. $3$-uniform states can
be constructed by forming bi-layer graphs with $n/2$ qubits ($n=2\mathbb{Z}$)
in each layer, such that each layer forms a fully connected graph while
inter-layer connections are such that the vertices in one layer has a one to
one connectivity to the other layer. $4$-uniform states can be formed by taking
2D lattice graphs( also referred elsewhere as a 2D cluster Ising state ) with
periodic boundary conditions along both dimensions and both dimensions having
at least $5$ vertices.
