We enumerate the 2-\((9,4,6)\) designs and find \(270,474,142\) non-isomorphic such designs in a backtrack search. The sizes of their automorphism groups vary between \(1\) and \(360\). Out of these designs, \(19,489,464\) are simple and \(2,148,676\) are decomposable.

A \(t\)-partite number is a \(t\)-tuple \(\vec{n} = (n_1, \ldots, n_t)\), where \(n_1, \ldots, n_t\) are positive integers. For a \(t\)-partite number \(\vec{n}\), let \(f_t(\vec{n})\) be the number of different ways to write \(\vec{n}\) as a product of \(t\)-partite numbers, where the multiplication is performed coordinate-wise, \((1, 1, \ldots, 1)\) is not used as a factor of \(\vec{n}\), and two factorizations are considered the same if they differ only in the order of the factors. This paper gives the following explicit upper bound for the multiplicative partition function \(f_t(\vec{n})\):
\[f_t(n_1, \ldots, n_t) \leq M^{w(t)},\, \text{where}\,\, M = \Pi_{i=1}^t n_i \,\,\text{and}\,\, w(t) = \frac{\log((t+1)1)}{t\log2}\].

The following partition problem was first introduced by R.C. Entringer and has subsequently been studied by the first author and more recently by Bollobas and Scott, who consider the hypergraph version as well, using a probabilistic technique. The partition problem is that of coloring the vertex set of a graph with \(s\) colors so that the number of induced edges is bounded for each color class. The techniques employed are non-constructive and non-probabilistic and improve the known bounds in the previous papers.

In a communication network, several vulnerability measures are used to determine the resistance of the network to disruption of operation after the failure of certain stations or communication links. If we think of a graph as modeling a network, the edge-integrity of a graph is one \(\textbf{measure of graph vulnerability}\) and it is defined to be the minimum sum of the orders of a set of edges being removed and a largest remaining component. In this paper, the edge-integrity of graphs \(B_n\), \(H_n\), and \(E_p^t\), are calculated. Also, some results are given about edge-integrity of these graphs.

In this paper, it is shown that the necessary condition for the existence of a holey perfect Mendelsohn design (HPMD) with block size 5, type \(h^n\) and index \(\lambda\), namely, \(n \geq 5\) and \(\lambda n(n-1)h^2 \equiv 0 \pmod{5}\), is also sufficient for \(\lambda \geq 2\). The result guarantees the analogous existence result for group divisible designs (GDDs) of type \(h^n\) having block size 5 and index \(4\lambda\).

The computational complexity of the graph isomorphism problem is still unknown. We consider Cartesian products \(K_n \times K_m\) of two complete graphs \(K_n\) and \(K_m\). An acyclic orientation of such a Cartesian product is called a sequence graph because it has an application in production scheduling. It can be shown that the graph isomorphism problem on the class of these acyclic digraphs is solvable in polynomial time. We give numbers of non-isomorphic sequence graphs for small \(n\) and \(m\). The orientation on the cliques of a sequence graph can be interpreted as job orders and machine orders of a shop scheduling problem with a complete operation set.

Tenacity is a recently introduced parameter to measure vulnerability of networks and graphs. We characterize graphs having the maximum number of edges among all graphs with a given number of vertices and tenacity.

In this paper, we show that some graphs are circuit unique by applying a new tool, which is the character of the matching polynomial. Some properties of the character of the matching polynomial is also given.

The theory of hypergeometric functions is brought to bear on a problem—namely, that of obtaining a certain power series expansion involving the sine function that is inclusive of the Catalan sequence and which serves as a prelude to the calculation of other related series of similar type. A general formulation provides the particular result of interest as a special case, into which Catalan numbers are introduced as desired.