  In geometric analysis, an index theorem relates the difference of the numbers
of solutions of two differential equations to the topological structure of the
manifold or bundle concerned, sometimes using the heat kernels of two
higher-order differential operators as an intermediary. In this paper, the case
of quantum graphs is addressed. A quantum graph is a graph considered as a
(singular) one-dimensional variety and equipped with a second-order
differential Hamiltonian H (a "Laplacian") with suitable conditions at
vertices. For the case of scale-invariant vertex conditions (i.e., conditions
that do not mix the values of functions and of their derivatives), the constant
term of the heat-kernel expansion is shown to be proportional to the trace of
the internal scattering matrix of the graph. This observation is placed into
the index-theory context by factoring the Laplacian into two first-order
operators, H =A*A, and relating the constant term to the index of A. An
independent consideration provides an index formula for any differential
operator on a finite quantum graph in terms of the vertex conditions. It is
found also that the algebraic multiplicity of 0 as a root of the secular
determinant of H is the sum of the nullities of A and A*.
