In the present paper, motivated by \cite{MR,SS}, first we give a notion of graphical $b_{v}(s)$-metric space, which is a graphical version of $b_{v}(s)$-metric space. Utilizing the graphical Banach contraction mapping we prove fixed point results in graphical $b_{v}(s)$-metric space. Appropriate examples are also presented to support our results. In the end, the main result ensures the existence of a solution for an ordinary differential equation along with its boundary conditions by using the fixed point result in graphical $b_{v}(s)$-metric space.