A direct method for constructing large sets of \(t\)-designs is based on the concept of assembling orbits of a permutation group \(G\) on \(k\)-subsets of a \(v\)-set into block sets of \(t\)-designs so that these designs form a large set. If \(G\) is \(t\)-homogeneous, then any orbit is a \(t\)-design and therefore we obtain a large set by partitioning the set of orbits into parts consisting of the same number of \(k\)-subsets. In general, it is hard to find such partitions. We solve this problem when orbit sizes are limited to two values. We then use its corollaries to obtain some results in a special case in which a simple divisibility condition holds and no knowledge about orbit sizes is assumed.

Dean \(([3])\) shows that if \(G\) be a \(k\)-connected graph such that any fragment whose neighborhood contains an edge has cardinality exceeding \(\frac{k}{2}\), then the subgraph \(H = (V(G), E_k(G))\) formed by \(V(G)\) and the \(k\)-contractible edges of \(G\) is \(2\)-connected. In this paper, we show that for \(k = 4\), Dean’s result holds when reduced \(\frac{k}{2}\) to \(\frac{k}{4}\). But for \(k \geq 5\), we give a counterexample to show that it is false and give a lower bound of the number of \(k\)-contractible edges for \(k = 5\).

Let \(G\) be a finite simple \(\chi\)-chromatic graph and \(\mathcal{L} = \{L_u\}_{u\in V(G)}\) be a list assignment to its vertices with \(L_u \subseteq \{1,…,\chi\}\). A list colouring problem \((G, \mathcal{L})\) with a unique solution for which the sum \(\sum_{u\in V(G)}|L_u|\) is maximized, is called a \(maximum\; \chi-list \;assignment\) of \(G\). In this paper, we prove a \(Circuit\; Simulation\) Lemma that, strictly speaking, makes it possible to simulate any Boolean function by \(effective\) 3-colourings of a graph that is \(polynomial-time \;constructable\) from the Boolean function itself. We use the lemma to simply prove some old results as corollaries, and also we prove that the following decision problem, related to the computation of the fixing number of a graph [Daneshgar \(1997\), Daneshgar and Naserasr, Ars Combin. \(69\) \((2003)\)], is \(\sum_{2}^{P}\)-complete.

In this paper, we give another proof for labeled spanning forests of the complete bipartite graph \(K_{m,n}\), and obtain two Abel-type polynomials. And then we investigate the enumeration of non-trivial rooted labeled spanning forests of the complete bipartite graph \(K_{m,n}\).

In this paper, we investigate the global behavior of the difference equation

\[x_{n+1} = \frac{\alpha x_{n-k}}{\beta+\gamma x_{n-(k+1)}^p},\text{n=1,2,}\ldots\]

with non-negative parameters and non-negative initial conditions, where \(k\) is an odd number.

In many papers, the relation between the domination number of a product of graphs and the product of domination numbers of factors is studied. Here we investigate this problem for domination and total domination numbers in the cross product of digraphs. We give analogues of known results for graphs, and we also present new results for digraphs with sources. Using these results, we find domination (total domination) numbers for some classes of digraphs.

Let \(P_{k+1}\) denote a path of length \(k\) and let \(C_k\) denote a cycle of length \(k\). As usual, \(K_n\) denotes the complete graph on \(n\) vertices. In this paper, we investigate decompositions of \(K_n\) into paths and cycles, and give some necessary and/or sufficient conditions for such a decomposition to exist. Besides, we obtain a necessary and sufficient condition for decomposing \(K_n\) into \(p\) copies of \(P_5\) and \(q\) copies of \(C_4\) for all possible values of \(p\geq 0\) and \(q\geq 0\).

Given a simple connected undirected graph \(G\), the Wiener index \(W(G)\) of \(G\) is defined as half the sum of the distances over all pairs of vertices of \(G\). In practice, \(G\) corresponds to what is known as the molecular graph of an organic compound. We obtain a sharp lower bound for \(W(G)\) of an arbitrary graph in terms of the order, size, and diameter of \(G\).

The Zagreb indices are topological indices of graphs, which are defined as:\(M_1(G) = \sum\limits_{v \in V(G)} (d(v))^2\), \(M_2(G) = \sum\limits_{uv \in E(G)} d(u)d(v)\) .In this paper, we determine the upper and lower bounds for the Zagreb indices of unicyclic graphs in terms of their order and girth. In each case, we characterize the extremal graphs.

In this paper, we are concerned with Leibniz numbers. We establish a series of identities involving Leibniz numbers, Stirling numbers, harmonic numbers, arctan numbers by making use of generating functions. In addition, we give the asymptotic expansion of certain sums related to Leibniz numbers by Laplace’s method.