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Melham conjectured two identities between Fibonacci and Lucas numbers in 1999. Subsequently, Kilic et al. showed them in 2010 by contour integration. In this paper, we present a new proof using the q -derivative operator. By examining two rational functions, we further derive several identities concerning Fibonomial coefficients. Finally, two well-known summation formulae on q -series are transformed into convolution formulae on Fibonomial coefficients. [ABSTRACT FROM AUTHOR]

Abstract: Let be the graph obtained from by adding a new vertex corresponding to each edge of and by joining each new vertex to the end vertices of the corresponding edge, and be the graph obtained from by inserting a new vertex into every edge of and by joining by edges those pairs of these new vertices which lie on adjacent edges of . In this paper, we determine the Laplacian polynomials of and of a regular graph ; on the other hand, we derive formulae and lower bounds of the Kirchhoff index of these graphs. [Copyright &y& Elsevier]

Abstract: This paper explores new connections between the satisfiability problem and semidefinite programming. We show how the process of resolution in satisfiability is equivalent to a linear transformation between the feasible sets of the relevant semidefinite programming problems. We call this transformation semidefinite programming resolution, and we demonstrate the potential of this novel concept by using it to obtain a direct proof of the exactness of the semidefinite formulation of satisfiability without applying Lasserre’s general theory for semidefinite relaxations of 0/1 problems. In particular, our proof explicitly shows how the exactness of the semidefinite formulation for any satisfiability formula can be interpreted as the implicit application of a finite sequence of resolution steps to verify whether the empty clause can be derived from the given formula. [Copyright &y& Elsevier]