The determinant of a unicyclic graph's neighborhood matrix (Q1765895)

Let \(G\) be a unicyclic graph with \(n\) vertices and a unique cycle, \(A(G)\) denotes the adjacency matrix of the graph \(G\). The algorithm for computing the determinant function of the matrix \(\alpha I_n+A(G)\) which uses \(O(n)\) arithmetic operations under some restrictions on the degrees of the vertices of the graph \(G\) is obtained. Among the corollaries from the above results the author points out the algorithms to compute the determinants of adjacency and neigborhood matrices and the characteristic polynomial for adjacency matrices.