For \(n\) a positive integer and \(v\) a vertex of a graph \(G\), the \(n\)th order degree of \(v\) in \(G\), denoted by \(\text{deg}_n(v)\), is the number of vertices at distance \(n\) from \(v\). The graph \(G\) is said to be \(n\)th order regular of degree \(k\) if, for every vertex \(v\) of \(G\), \(\text{deg}_n(v) = k\). For \(n \in \{7, 8, \ldots, 11\}\), a characterization of \(n\)th order regular trees of degree \(2\) is obtained. It is shown that, for \(n \geq 2\) and \(k \in \{3, 4, 5\}\), if \(G\) is an \(n\)th order regular tree of degree \(k\), then \(G\) has diameter \(2n – 1\).

We prove that there exist precisely \(459\) pairwise non-isomorphic Steiner systems \(S(5,6,48)\) stabilized by the group \({PSL}_2(47)\).

The known generalized quadrangles with parameters \((s,t)\) where \(|s-t| = 2\) have been characterized in several ways by M. De Soete \([D]\), M. De Soete and J. A. Thas \([DT1]\), \([DT2]\), \([DT4]\), and the present author \([P]\). Certain of these results are interpreted for a coset geometry construction.

In this paper, we illustrate the relationship between profiles of Hadamard matrices and weight distributions of codes, give a new and efficient method to determine the minimum weight \(d\) of doubly even self-dual \([2n,n,d]\) codes constructed by using Hadamard matrices of order \(n = 8t + 4\) with \(t \geq 1\), and present a new proof that the \([2n,n,d]\) codes have \(d \geq 8\) for all types of Hadamard matrices of order \(n = 8t + 4\) with \(t \geq 1\). Finally, we discuss doubly even self-dual \([72,36,d]\) codes with \(d = 8\) or \(d = 12\) constructed by using all currently known Hadamard matrices of order \(n = 36\).

We define an \({extremal \; graph}\) on \(v\) vertices to be a graph that has the maximum number of edges on \(v\) vertices, and that contains neither \(3\)-cycles nor \(4\)-cycles.
We establish that every vertex of degree at least \(3\), in an extremal graph of at least \(7\) vertices, is in a \(5\)-cycle; we enumerate all of the extremal graphs on \(21\) or fewer vertices; and we determine the size of extremal graphs of orders \(25\), \(26\), and \(27\).

We consider square arrays of numbers \(\{a(n, k)\}\), generalizing the binomial coefficients:
\(a(n, 0) = c_n\), where the \(c_n\) are non-negative real numbers; \(a(0, k) = c_0\), and if \(n, k > 0\), then \(a(n, k) = a(n, k – 1) + a(n – 1, k)\).
We give generating functions and arithmetical relations for these numbers. We show that every row of such an array is eventually log concave, and give a few sufficient conditions for columns to be eventually log concave. We also give a necessary condition for a column to be eventually log concave, and provide examples to show that there exist such arrays in which no column is eventually log concave.

In this paper, we obtain some necessary conditions for the existence of balanced arrays (\(B\)-arrays) of strength \(4\) and with two levels, and we state the usefulness of these conditions in obtaining an upper bound on the number of constraints for these B-arrays.

It is shown that the circuit polynomial of a graph, when weighted by the number of nodes in the circuits, does not characterize the graph, i.e., non-isomorphic graphs can have the same circuit polynomial. Some general theorems are given for constructing graphs with the same circuit polynomial (cocircuit graphs). Analogous results can be deduced for characteristic polynomials.

Given \(n\) real numbers whose sum is zero, find one of the numbers that is non-negative. In the model under consideration, an algorithm is allowed to compute \(p\) linear forms in each time step until it knows an answer. We prove that exactly \(\lceil{\log n}/{\log(p+1)} \rceil\) time steps are required. Some connections with parallel group-testing problems are pointed out.

With the help of a computer, the third Ramsey number is determined for each of the \(25\) graphs with five edges, five or more vertices, and no trivial components.