The determinant of a unicyclic graph's neighborhood matrix

Let G be a unicyclic graph with n vertices and the unique cycle C, A(G) and N(G) its adjacency matrix and neighborhood matrix, respectively, and α a scalar. We obtain an algorithm for computing the determinant of α In + A(G) which uses O(n) space and O(n) arithmetic operations for δ⩽n, where δ = min{dG(x) : x ∈ V(C)}. Applications include computing the determinants of A(G) and N(G), and computing the characteristic polynomial of A(G).