  We present a systematic collection of spectral surgery principles for the
Laplacian on a metric graph with any of the usual vertex conditions (natural,
Dirichlet or $\delta$-type), which show how various types of changes of a local
or localised nature to a graph impact the spectrum of the Laplacian. Many of
these principles are entirely new, these include "transplantation" of volume
within a graph based on the behaviour of its eigenfunctions, as well as
"unfolding" of local cycles and pendants. In other cases we establish sharp
generalisations, extensions and refinements of known eigenvalue inequalities
resulting from graph modification, such as vertex gluing, adjustment of vertex
conditions and introducing new pendant subgraphs.
  To illustrate our techniques we derive a new eigenvalue estimate which uses
the size of the doubly connected part of a metric graph to estimate the
spectral gap. This quantitative isoperimetric-type inequality interpolates
between two known estimates---one assuming the entire graph is doubly connected
and the other making no connectivity assumption (and producing a weaker
bound)---and includes them as special cases.
