J. Leech has posed the following problem: For each integer \(n\), what is the greatest integer \(N\) such that there exists a labelled tree with \(n\) nodes in which the distance between the pairs of nodes include the consecutive values \(1,2,\ldots,N\)? With the help of a computer, we get \(B(n)\) (the number \(N\) for branched trees) for \(2 \leq n \leq 10\) and lower bounds of \(B(11)\) and \(B(12)\). We also get \(U(n)\) (the number \(N\) for unbranched trees) for \(2 \leq n \leq 11\) independently, confirming some results gotten by J. Leech.

A method is presented for constructing simple partially balanced designs from \(t-(v,k,\lambda)\) designs. When the component designs satisfy a compatibility condition the result is a simple balanced design. The component designs can even be trivial (with some exceptions) with the resulting design being nontrivial. The automorphism group of the composition is given in terms of the automorphism groups of the component designs. Some previously unknown simple designs are constructed, including an infinite family of \(3\)-designs that are extremal with respect to an inequality of Cameron and Praeger. Some analogous theorems are given for difference families.

In this paper, constructions of simple cyclic \(2\)-designs are given. As a consequence, we determined the existence of simple \(2\)-\((q,k,\lambda)\) designs for every admissible parameter set \((q,k,\lambda)\) where \(q \leq 29\) is an odd prime power, with two undecided parameter sets \((q,k,\lambda) = (29,8,6)\) and \((29,8,10)\).

A map is an embedding of a graph into a surface so that each face is simply connected. Geometric duality, whereby vertices and faces are reversed, is a classic construction for maps. A generalization of map duality is given and discussed both graph and group theoretically.

We show how a claw-free well-covered graph containing no \(4\)-cycle, with any given independence number \(m\), can be constructed by linking together \(m\) sub-graphs, each isomorphic to either \(K_2\) or \(K_3\). We show further that the only well-covered connected claw-free graphs containing no \(4\)-cycle that cannot be constructed in this way are \(K_1\), and the cycle graphs on \(5\) and \(7\) vertices respectively.

In [3] R. Brauer asked the question: When is an \(n \times n\) complex matrix \(X\) the ordinary character table of some finite group? It is shown that the problem can be reduced in polynomial time to that of VERTEX INDEPENDENCE. We also pose and solve some (much) simpler problems of a related combinatorial nature.

Let \(\mathbb{F}_q = \text{GF}(q^n)\) denote the finite field of order \(q^n\), and let \(U_n = \cup_{i=1}^{n}(\mathbb{F}_i,)\). Explicit permutation-type formulas for the Frobenius map \(\varphi\) (defined by \((\varphi(x)) = x^q\)) on \(\mathbb{F}_n\) and on \(U_n\) are obtained by using the well-known number \(N_q\) (the number of monic irreducible polynomials of degree \(i\) in \(\mathbb{F}_1[x]\)). Some results in [1] and [2] can be obtained from these formulas. Moreover, some other results are also given by using Pólya’s counting theory.

The composition of two graphs \(G\) and \(H\), written \(G[H]\), is the graph with vertex set \(V(G) \times V(H)\) and \((u_1,v_1)\) is adjacent to \((u_2,v_2)\) if \(u_1\) is adjacent to \(u_2\) in \(G\) or if \(u_1 = u_2\) and \(v_1\) is adjacent to \(v_2\) in \(H\). The \(r\)th power of graph \(G\), denoted \(G^r\), is the graph with vertex set \(V(G)\) and edge set \(\{(u,v) : d(u,v) \leq r \text{ in } G\}\). The bandwidth of graph \(G\) is \(\min \max |f(u) – f(v)|\), where the max is taken over each edge \(uv \in E(G)\), and the min is over all proper numberings \(f\). This paper establishes tight upper and lower bounds for the bandwidth of an arbitrary graph composition \(G[H]\), with the upper bound based only on \(|V(H)|\) and the bandwidth of \(G\). In addition, the exact bandwidth of the composition of \(G[H]\) is established for \(G\) the power of a path or a cycle.

The edge-neighbor-connectivity of a graph \(G\) is the minimum size of all edge-cut-strategies of \(G\), where an edge-cut-strategy consists of a set of common edges of double stars whose removal disconnects the graph \(G\) or leaves a single vertex or \(\emptyset\). This paper discusses the extreme values of the edge-neighbor-connectivity of graphs relative to the connectivity, \(\kappa\), and gives two classes of graphs — one class with minimum edge-neighbor-connectivity, and the other one with maximum edge-neighbor-connectivity.