The authors study the spectrum of the continuous Laplacian \(\Delta\) on a countable connected locally finite graph \(\Gamma\) without self-loops. They assume the edges of the graph have suitable positive conductances and are identified with copies of line segments \([0,1]\). They also assume the condition that the sum of the weighted normal exterior derivatives is \(0\) at every node. This is the Kirchhoff-type condition. Let \(P\) be the transition operator on the vertex set. The authors study the relationship between the \(L^2\) spectrum of the operator \(\Delta\) and the \(\ell^2\) spectrum of the discrete Laplacian \(I-P\).