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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.

In this paper we revisit the seminal coercion-resistant e-voting protocol by Juels, Catalano and Jakobsson (JCJ) and in particular the attempts to make it usable and practical. In JCJ the user needs to handle cryptographic credentials and be able to fake these in case of coercion. In a series of three papers Neumann et al. analysed the usability of JCJ, and constructed and implemented a practical credential handling system using a smart card which unlock the true credential via a PIN code, respectively fake the credential via faking the PIN. We present several attacks and problems with the security of this protocol, especially an attack on coercion-resistance due to information leakage from the removal of duplicate ballots.

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.

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.