arXiv2026
Let $H$ be a finite simple graph. A one-particle continuous-time quantum walk on its line graph $\ell H$ has one mode for each edge of $H$. This paper is a corrected and reorganized account of that setting. We first record the matrix encoding of an edge-amplitude vector, its Schur state. The encoding is necessarily complex symmetric if it is to be complex linear and to retain the global phase, and it is normalized by $1/\sqrt{2}$. It is unitary for the Frobenius inner product and is closed under a canonical tensor lift to the categorical product of graphs. The tensor lift also pins down the convention. Among all unimodular phase variants the symmetric choice is the only one that survives, and every relaxed variant reduces to a sign. We then replace the labelled tensor-power description by the bosonic and fermionic Fock spaces over $\ell^2(EH)$, and assemble the standard free-particle apparatus in the form used afterwards. This covers the occupation-number basis, the second-quantized line-graph Hamiltonian, permanent and determinant formulae for many-particle transition amplitudes, and the one-particle reduced density matrix. The one graph-theoretic input is the even-Eulerian criterion, which is the zero-sum $2$-flow criterion of Wang and Hu, recalled here with a short proof to fix conventions. Its translation through the incidence identity gives a real equimodular full-support $-2$ eigenmode of $A(\ell H)$, and hence bosonic condensates of energy $-2N$ that are uniform over the edges of $H$. The fermionic case is stated separately so that the Pauli constraint is explicit.