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Let Γ be a graph with n vertices, where each edge is given an orientation and let Q be the vertex–edge incidence matrix of Γ . Suppose that Γ has a cut-vertex v and Γ − v = Γ [ V 1 ] ∪ Γ [ V 2 ] . We obtain a relation between the Moore–Penrose inverse of the incidence matrix of Γ and of the incidence matrices of the induced subgraphs Γ [ V 1 ∪ { v } ] and Γ [ V 2 ∪ { 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.

The distinguishing number D(G) of a graph G is the minimum number of colors needed to color the vertices of G such that the coloring is preserved only by the trivial automorphism. In this paper we improve results about the distinguishing number of Cartesian products of finite and infinite graphs by removing restrictions to prime or relatively prime factors.