Moore-Penrose inverse of incidence matrix of graphs with complete and cyclic blocks

Let Gamma be a graph with n vertices, where each edge is given an orientation and let Q be the vertex-edge incidence matrix of Gamma. Suppose that Gamma has a cut-vertex v and Gamma-v = Gamma[V-1]boolean OR Gamma[V-2]. We obtain a relation between the Moore-Penrose inverse of the incidence matrix of Gamma and of the incidence matrices of the induced subgraphs Gamma[V-1 boolean OR {v}] and Gamma[V-2 boolean OR {v}]. The result is used to give a combinatorial interpretation of the Moore-Penrose inverse of the incidence matrix of a graph whose blocks are either cliques or cycles. Moreover we obtain a description of minors of the Moore-Penrose inverse of the incidence matrix when the rows are indexed by cut-edges. The results generalize corresponding results for trees in the literature. (C) 2018 Elsevier B.V. All rights reserved.