In this paper, we show that for all v ≡ 1 (mod 3), there exists a supersimple (v, 4, 2) directed design. Also, we show that for these parameters there exists a super-simple (v, 4, 2) directed design whose each defining set has at least a half of the blocks.

A generalized-theta-graph is a graph consisting of a pair of end vertices joined by k (k ≥ 3) internally disjoint paths. We denote the family of all the n-vertex generalized-theta-graphs with k paths between end vertices by Θ n k . In this paper, we determine the sharp lower bound and the sharp upper bound for the total number of matchings of generalized-theta-graphs in Θ n k . In addition, we characterize the graphs in this class of graphs with respect to the mentioned bounds.

In this paper, we determine the tight upper bound for the number of matchings of connected $n$-vertex tricyclic graphs. We show that this bound is $13 f_{n-4} + 16f_{n-5}$, where $f_n$ be the  $n$th Fibonacci number. We also  characterize the $n$-vertex simple connected tricyclic graph for which the bound is best  possible.A corrigendum was added to this paper on Jun 17, 2015.

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