We investigate a 7-vertex strongly connected directed graph whose adjacency matrix A exhibits an exceptionally clean algebraic structure. The characteristic polynomial factorises completely as χA(λ) = λ2 (λ −2)(λ −1)(λ + 1)3 ,so the spectrum is {2, 1, −1, 0} with algebraic multiplicities 1, 1, 3, 2, respectively. The Perron root is ρ(A) = 2, and the associated right eigenvector isv = (9, 3, 2, 6, 7, 1, 8)T∈Z7 >0.For every directed edge i →j we define the rational weight wij = vj/(2vi). These weights form a row-stochastic matrix. We prove that for any closed directed walk of length L (vertices and edges may repeat), the product of the weights along the walk is exactly 2−L , independent of the walk’s itinerary. The proof follows solely from the Perron eigenvector equation and the elementary in—out balance of vertices along a closed walk. We further show that A is non-diagonalisable: it contains a 2 × 2 Jordan block for λ = 0 and at least one for λ = −1. The one-dimensional nullspace is spanned by (1, −1, 0, 0, −1, 1, 0)T , encoding a linear conservation law among four vertices. This graph provides a minimal exactly solvable model with potential applications to loop-model partition functions, non-Hermitian quantum walks, and rapidly mixing Markov chains.Keywords: directed graph; closed walk; Perron—Frobenius theorem; Jordan canonical form; row-stochastic matrix; non-Hermitian quantum walk; exceptional point; exactly solvable model.