together with the existence of an exact trace formula, has lead to the suggestion that their study might provide insights into some of the fundamental problems of quantum chaos [@kot:2]. This has motivated many works considering a variety of aspects of quantum graphs, [@bar:lsd], [@berk:3], [@bolte:1], [@desbois:1], [@kurasov:1], [@prot:1], [@pascaud:1], [@schanz:1], [@schanz:2], [@tanner:1].

Studies [@berk:2], [@berk:1] of a special class of graphs - the so-called “hydra” graphs or star graphs - have revealed spectral statistics that are not typically associated with quantum chaotic systems. These have been dubbed “intermediate statistics” in recent works, [@bogomolny:1], [@bogomolny:3], [@bogomolny:2] and have been observed in a number of systems. We are motivated to investigate this further by studying the value distributions of the eigenfunctions and the spectral determinant of the Schrödinger operator on quantum star graphs.

A [*star graph*]{} consists of a single central vertex together with $v$ outlying vertices each of which is connected only to the central vertex by a bond (figure \[fig:0\]). Hence there are $v$ bonds. We associate to each bond a length $L_j, j=1,\ldots,v$. We will often refer to the vector of bond lengths ${{\mathbf L}}:=(L_1,\ldots,L_v)$.

![A star graph with 5 bonds[]{data-label="fig:0"}](star.eps){width="3.0cm" height="2.5cm"}

The Schrödinger operator on a star 