  The paper contains the probabilistic analysis of the Brownian motion on the
simplest quantum graph, spider: a system of N-half axis connected only at the
graph's origin by the simplest (so-called Kirchhoff's) gluing conditions. The
limit theorems for the diffusion on such a graph, especially if $N \to \infty$
are significantly different from the classical case $N = 2$ (full axis).
Additional results concern the properties of the spectral measure of the spider
Laplacian and the corresponding generalized Fourier transforms. The
continuation of the paper will contain the study of the spectrum for the class
of Schr\"odinger operators on the spider graphs: Laplacian perturbed by
unbounded potential and related phase transitions.
