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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 Wiener index is a topological index of a molecule, defined as the sum of distances between all pairs of vertices in the chemical graph representing the non-hydrogen atoms in the molecule. Hexagonal chains consist of hexagonal rings connected with each other by edges. This class of graphs contains molecular graphs of unbranched catacondensed benzenoid hydrocarbons. A segment of a chain is its maximal subchain with linear connected hexagons. Chains with segments of equal lengths can be coded by binary words. Formulas for the sums of Wiener indices of hexagonal chains of some families are derived and computational examples are presented.

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.

Let G be a connected graph of order n having ɛ ( G ) end-vertices. Given a positive integer t , we denote by S ( G , t ) the t -th generalized Sierpinski graph of G . In this note we show that if every internal vertex of G is a cut vertex, then the strong metric dimension of S ( G , t ) is given by dim s ( S ( G , t )) = ( ɛ ( G )( n t − 2 n t − 1 + 1) − n + 1) / ( n − 1) .

For a finite bounded lattice £, we associate a zero-divisor graph G(£) which is a natural generalization of the concept of zero-divisor graph for a Boolean algebra. Also, we study the interplay of lattice-theoretic properties of £ with graph-theoretic properties of G(£).

In this paper, using module endomorphisms, we extend the concept of the zero-divisor graph of a ring to a module over an arbitrary commutative ring. The main aim of this article is studying the interplay of module-theoretic properties of a module with graph properties of its zero-divisor graph.