Volume 34, Issue 5

We consider a hyperplane arrangement constructed from a subset of the set of all simple paths in a graph. A relation of the constructed arrangement to the maximum matching problem is established. In addition, the problem of finding the characteristic polynomial is reduced to the case of a connected original graph. In the case where the original graph is a tree, a formula for the characteristic polynomial is obtained.

We consider representations of an arbitrary permutation π of degree 2 n , n ⩾ 3, by products of the so-called (2 n )-permutations (any cycle of such a permutation has length 2). We show that any even permutation is represented by the product of four (2 n )-permutations. Products of three (2 n )-permutations cannot represent all even permutations. Any odd permutation is realized (for odd n ) by a product of five (2 n )-permutations.

We consider Kuznetsov’s implicit expressibility and its generalizations, when the implicit expressibility language is augmented with the additional disjunction, implication, and negation logical connectives. It is shown that, for each k ⩾ 3, the implicit extensions in P k have the cardinality of the continuum. For each k ⩾ 3, we also prove that each of the sets of positively implicit, implicatively implicit, and negatively implicit extensions in P k contains, respectively, as a proper subset, the set of positively implicit, implicatively implicit, and negatively implicit closed classes. We verify that, for k ⩾ 2, the functions of the set Hk∗ $\begin{array}{} \displaystyle H_k^* \end{array}$ of homogeneous functions preserving the set E k −1 can be used for producing implicatively implicit and negatively implicit extensions without changing the result.

We show that Hamming graphs of diameter D are radio graceful provided that each component of the Hamming graph is a complete graph with at least D vertices. This generalizes the results known previously for diameters D = 2 and D = 3 and for general D with the assumption that the components of the Hamming graph have the same or pairwise relatively prime numbers of vertices.

For a sequence consisting of independent random variables having a Bernoulli distribution with the parameter p = 12 $\begin{array}{} \displaystyle \frac12 \end{array}$ the limit joint distribution of the statistics T 1 , T 2 , T 3 of the following three tests of the NIST package is obtained: «Monobit Test», «Frequency Test within a Block»and «Test for the Longest Run of Ones in a Block». It is proved that the covariance matrix C of the limit distribution of the vector ( T 1 , T 2 , T 3 ) satisfies the relations C 12 = C 21 = C 13 = C 31 = 0, C 23 = C 32 ≥ 0. For arbitrary p necessary and sufficient conditions for asymptotic uncorrelatedness and/or asymptotic independence of these statistics are obtained. The limit behavior of the vector ( T 1 , T 2 , T 3 ) is described for a wide class of values p ≠ 12 $\begin{array}{} \displaystyle \frac12 \end{array}$.

Let G be a connected graph. The resistance distance between any two vertices of G is defined to be the network effective resistance between them if each edge of G is replaced by a unit resistor. The Kirchhoff index of G is the sum of resistance distances between all pairs of vertices of G . In this paper, we determine the resistance distance and Kirchhoff index of the subdivision double join G S ∨ { G 1 , G 2 } and R -graph double join G R ∨ { G 1 , G 2 } for a regular graph G and two arbitrary graphs G 1 , G 2 , respectively.