Betti numbers of normal edge rings (II)

We compute the Betti numbers of the edge rings of multi-path graphs using the \emph{induced-subgraph approach} introduced in \cite{WL1}. Here, a multi-path graph refers to a simple graph composed of several paths that have the same starting point and the same ending point. Special cases include the graph $G_{r,d}$ introduced in \cite{GHK}, the graph $G_{r,s,d}$ introduced in \cite{NN}, and the graph $B_{\underset{‾}{\mathrm{\ell }},h}$ introduced in \cite{LZ}. In particular, we show that all the multi-graded Betti numbers of a multi-path graph are the top multi-graded Betti numbers of some of its induced subgraphs.