Signed scattering-delay balance and exact nodal laws for adjacent edge-disjoint two-cycle pumpkin chains
Year: 2026
Type: preprint
Abstract: We determine the generic Barra--Gaspard nodal-surplus law of the standard quantum graph given by the pumpkin chain $[1,2,2,1]$. Its two cycles are edge-disjoint but meet at a vertex, so the graph lies outside the vertex-disjoint-cycle hypothesis under which binomial nodal statistics were previously proved. Splitting the graph at its central vertex reduces the dependence of its local surpluses to a scalar correlation on the secular manifold. For the resulting one-channel block $B=[1,2]$, we calculate the scattering phase explicitly and prove, by a fiberwise argument, that every edge-resolved phase-delay density weighted by the signed local surplus vanishes identically. We call this property signed scattering-delay balance. The Barra--Gaspard measure then makes the two local surpluses of $[1,2,2,1]$ independent symmetric Bernoulli variables, and the total surplus has the exact law $\operatorname{Bin}(2,1/2)$. For rationally independent edge lengths, the frequencies of generic eigenpairs with each surplus value, normalized by the full nonzero spectral counting function, converge to this law. The same calculation gives a quantitative contrast for the one-tailed chain $[1,2,2]$. If $L_B$ is the total length of its tailed block and $L_\circ$ the length of its bare-loop block, then the generic local-surplus correlation is $-\eta L_\circ/(2L_B+L_\circ)$, where $\eta=0.6368\ldots$. Together with the figure-eight graph, these results determine the generic nodal law of all three binary two-cycle pumpkin chains with adjacent edge-disjoint cycles.
Author: Nicolas, Jérôme
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7208765946