  We improve the worst-case information theoretic lower bound of Munro and Wu
(ISAAC 2018) for $n-$vertex unlabeled chordal graphs when vertex leafage is
bounded and leafage is unbounded. The class of unlabeled $k-$vertex leafage
chordal graphs that consists of all chordal graphs with vertex leafage at most
$k$ and unbounded leafage, denoted $\mathcal{G}_k$, is introduced for the first
time. For $k>0$ in $o(n/\log n)$, we obtain a lower bound of $((k-1)n \log n
-kn \log k - O(\log n))-$bits on the size of any data structure that encodes a
graph in $\mathcal{G}_k$. Further, for every $k-$vertex leafage chordal graph
$G$ such that for $k>1$ in $o(n^c), c >0$, we present a $((k-1)n \log n + o(kn
\log n))-$bit succinct data structure, constructed using the succinct data
structure for path graphs with $kn/2$ vertices. Our data structure supports
adjacency query in $O(k \log n)$ time and using additional $2n \log n$ bits, an
$O(k^2 d_v \log n + \log^2 n)$ time neighbourhood query where $d_v$ is degree
of $v \in V$.
