we find recursive formulae for the number of perfect matchings in a graph g by splitting g into subgraphs h and q. we use these formulas to count perfect matching of p hypercube qn. we also apply our formulas to prove that the number of perfect matching in an edge-transitive graph is , where denotes the number of perfect matchings in g, is the graph constructed from by deleting edges with an en...

The study of perfect state transfer on graphs has attracted a great deal attention during the past ten years because its applications to quantum information processing and computation. Perfect is understood be rare phenomenon. This paper establishes necessary sufficient conditions for bi-Cayley graph having over any given finite abelian group. As corollaries, many known new results are obtained...

We find recursive formulae for the number of perfect matchings in a graph G by splitting G into subgraphs H and Q. We use these formulas to count perfect matching of P hypercube Qn. We also apply our formulas to prove that the number of perfect matching in an edge-transitive graph is , where denotes the number of perfect matchings in G, is the graph constructed from by deleting edges with an en...

Imagine the situation: “One day, you notice an old country woman crossing a downtown street, just about to put one foot down on the rails of the streetcar line. [...] Suppose, now, that at the very moment she puts her foot on the rail a streetcar comes rushing down the tracks toward her. If the old woman does not notice the car and continues across the tracks nothing will happen. But if someone...

In the last decade there have been many results about special families of graphs whose number of perfect matchings is given by perfect or near perfect powers (N. Elkies et al., J. Algebraic Combin. 1 (1992), 111– 132; B.-Y. Yang, Ph.D. thesis, Department of Mathematics, MIT, Cambridge, MA, 1991; J. Propp, New Perspectives in Geometric Combinatorics, Cambridge University Press, 1999). In this pa...

In 1961, P.W. Kasteleyn enumerated the domino tilings of a 2n × 2n chessboard. His answer was always a square or double a square (we call such a number "squarish"), but he did not provide a combinatorial explanation for this. In the present thesis, we prove by mostly combinatorial arguments that the number of matchings of a large class of graphs with 4-fold rotational symmetry is squarish; our ...

We construct non-interactive zero-knowledge arguments for circuit satisfiability and arithmetic circuits with perfect completeness, perfect zero-knowledge and computational (co-)soundness. The non-interactive zeroknowledge arguments have sub-linear size and very efficient public verification. Our construction uses bilinear groups and is only proven secure in the generic group model, but does no...