Reeections on the First Eigenvalue

The Laplacian is the second-order operator on functions given by (f) = ?div(grad(f)): As such, it is elliptic, self-adjoint, and is positive semi-deenite. It therefore has an L 2 basis of eigenfunctions and eigenvalues| that is, there are functions fg i g and non-negative numbers f i g such that 1 The collection of f i g's is called the spectrum of. The original interest in the Laplacian and its spectrum came from relations between mathematics and physics, in particular Joseph Fourier's rather penetrating analysis of the ow of heat. It is not hard to understand this connection. If we think of heat ow as being described by a bunch of particles whose distribution is described by a function f t , then the statement that heat moves from the hot points to the cold points gets translated to the phrase that the vector eld describing this ow is just ?grad(f t), so that the in-nitesimal change in f t is given by div(grad(f t)) (there is a confusing change of sign here, because divergence is positive on an outgoing ow, that is, when f t is decreasing). From this, it follows that the equation describing heat ow is given by @ @t (f t) + (f t) = 0: Using this, we can formally construct a \Heat Kernel" with the property that f t (x) = Z H t (x; y)f 0 (y) dy is the unique solution to the Heat Equation with initial conditions f 0 (x). A similar discussion leads to the Wave Equation @ 2 @t 2 (f t) + (f t) = 0: as the equation governing the vibrations which create sound. All of the above discussion was carried out without much reference to what kind of space it was carried out on. One 2 typically thinks of the \background space" as being a Rie-mannian manifold. But one may equally well take the \back-ground space" to be a graph. Here, a function f will denote a function on the vertices, with grad(f) a function on edges given by grad(f)(e) = f(y) ? f(x); where x and y are the endpoints of e, and div(V)(x) = 1 d(x) X x"e V (e); where d(x) denotes the number of vertices incident to x, so that (f)(x) = 1 d(x) X yx f(x) ? f(y); where \y x" means \there is an edge joining x to y." Actually, …