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We get a formula for the number of hamiltonian circuits of a graph through which we characterize the special hamiltonian graphs, that is containing a dominant circuit of minimal length. (2000): 05C45. Mathematics Subject Classification

We transfer the whole geometry of PG(3, q) over a non-singular quadric Q 4,q of PG(4, q) mapping suitably PG(3, q) over Q 4,q . More precisely the points of PG(3, q) are the lines of Q 4,q ; the lines of PG(3, q) are the tangent cones of Q 4,q and the reguli of the hyperbolic quadrics hyperplane section of Q 4,q . A plane of PG(3, q) is the set of lines of Q 4,q meeting a fixed line of Q 4,q . We remark that this representation is valid also for a projective space P 3,K over any field K and we apply the above representation to construct maximal partial spreads F in PG (3, q). For q even we get new cardinalities for F. For q odd the cardinalities are partially known. (2000): 51E20, 51E23. Mathematics Subject Classification

We prove some necessary and easily verifiable conditions for a graph to be Hamiltonian, in terms of easily constructible matrices. The interest of this research derives from the non-existence till now of so friendly conditions. (2000): 05C45. Mathematics Subject Classification

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