Every labeling of the vertices of a graph with distinct natural numbers induces a natural labeling of its edges: the label of an edge \( (x,y) \) is the absolute value of the difference of the labels of \( x \) and \( y \). By analogy with graceful labelings, we say that a labeling of the vertices of a graph of order \( n \) is minimally \( k \)-equitable if the vertices are labelled with \( 1, 2, \ldots, n \) and in the induced labeling of its edges every label either occurs exactly \( k \) times or does not occur at all. For \( m \geq 3 \), let \( C_m’ \) (denoted also in the literature by \( C_m \circ K_1 \) and called a corona graph) be a graph with \( 2m \) vertices such that there is a partition of them into sets \( U \) and \( V \) of cardinality \( m \), with the property that \( U \) spans a cycle, \( V \) is independent and the edges joining \( U \) to \( V \) form a matching. Let \( \mathcal{P} \) be the set of all pairs \( (m, k) \) of positive integers such that \( k \) is a proper divisor of \( 2m \) (i.e., a divisor different from \( 2m \) and \( 1 \)) and \( k \) is odd if \( m \) is odd. We show that \( C_m’ \) is minimally \( k \)-equitable if and only if \( (m,k) \in \mathcal{P} \).

We show that the number of points at distance \( i \) from a given point \( x \) in a dense near polygon only depends on \( i \) and not on the point \( x \). We give a number of easy corollaries of this result. Subsequently, we look to the case of dense near polygons \( S \) with an order in which there are two possibilities for \( t_Q \), where \( Q \) is a quad of \( S \), and three possibilities for \( (t_H, v_H) \), where \( H \) is a hex of \( S \). Using the above-mentioned results, we will show that the number of quads of each type through a point is constant. We will also show that the number of hexes of each type through a point is constant if a certain matrix is nonsingular. If each hex is a regular near hexagon, a glued near hexagon or a product near hexagon, then that matrix turns out to be nonsingular in all but one of the eight possible cases. For the exceptional case, however, we provide an example of a near polygon that does not have a constant number of hexes of each type through each point.

In the Euclidean plane, let \( A \), \( B \), \( C \) be noncollinear points and \( T \) be the union of the lines \( AB \), \( BC \), \( CA \). It is shown that there is a point \( P \) such that if \( \tilde{T} \) is the image of \( T \) by any nonrotating uniform expansion about \( P \), then \( T \cap \tilde{T} \) is generally a six-point set that lies on a circle.

We show that for each positive integer \( t \), for which there is a skew-type Hadamard matrix of order \( 4t \), there is a quasi-symmetric \( ((4t – 1)^2, (4t – 1)(2t – 1), t(4t – 3)) \) design.

The Moore upper bound for the order \( n(\Delta, 2) \) of graphs with maximum degree \( \Delta \) and diameter two is \( n(\Delta, 2) < \Delta^2 + 1 \). The only general lower bound for vertex symmetric graphs is \( n_{vt}(\Delta, 2) \geq \left\lfloor \frac{\Delta + 2}{2} \right\rfloor \left\lceil \frac{\Delta + 2}{2} \right\rceil \). Recently, a construction of vertex transitive graphs of diameter two, based on voltage graphs, with order \( \frac{8}{9} \left( \Delta + \frac{1}{2} \right)^2 \) has been given in [5] for \( \Delta = \frac{3q – 1}{2} \) and \( q \) a prime power congruent with 1 mod 4. We give an alternative geometric construction which provides vertex transitive graphs with the same parameters and, when \( q \) is a prime power not congruent to 1 modulo 4, it gives vertex transitive graphs of diameter two and order \( \frac{1}{2} (\Delta + 1)^2 \), where \( \Delta = 2q – 1 \). For \( q = 4 \), we obtain a vertex transitive graph of degree 6 and order 32.

We present an optimal algorithm to label the edges of a complete graph with integer lengths so that every Hamilton cycle has the same length. The algorithm is complete in the sense that every edge-labelling with this property is the output labelling of some run of this algorithm. Such edge-labellings are induced by half-integer vertex-labellings by adding the vertex labels on an edge’s ends to determine its label. The Fibonacci sequence arises in this connection.

Two players are presented with a finite, simple graph \( G = (V, E) \) that has no isolated vertices. They take turns deleting an edge from the graph in such a way that no isolated vertex is created. The winner is the last player able to remove an edge. We analyze this game when the graph \(G\) is a path of arbitrary length. In addition, some observations are made in the situation that the graph has an automorphism of a special type.

A (previously reported) surprising and attractive hypergeometric identity is established from first principles using three hypergeometric transformations.

Computational Algebra methods have been used successfully in various problems in many fields of Mathematics. Computational Algebra encompasses a set of powerful algorithms for studying ideals in polynomial rings and solving systems of nonlinear polynomial equations efficiently. The theory of Gröbner bases is a cornerstone of Computational Algebra, since it provides us with a constructive way of computing a kind of particular basis of an ideal which enjoys some important properties. In this paper, we introduce the concept of Hadamard ideals in order to establish a new approach to the construction of Hadamard matrices with circulant core. Hadamard ideals reveal the rich interplay between Hadamard matrices with circulant core and ideals in multivariate polynomial rings. Hadamard ideals yield an exhaustive search for Hadamard matrices with circulant core for any specific dimension. In particular, we furnish all solutions for Hadamard matrices of the 12 orders 4, 8, \ldots, 44, 48 with circulant core. We establish the dihedral structure of the varieties associated with Hadamard ideals. Finally, we furnish the complete lists (exhaustive search) of inequivalent Hadamard matrices of the 12 orders 4, 8, \ldots, 44, 48 with circulant core.

Let \( K_v \) be the complete graph on \( v \) vertices, and \( C_5 \) be a cycle of length five. A simple minimum \( (v, C_5, 1) \)-covering is a pair \( (V, C) \) where \( V = V(K_v) \) and \( C \) is a family of edge-disjoint 5-cycles of minimum cardinality which partition \( E(K_v) \cup E \), for some \( E \subset E(K_v) \). The collection of edges \( E \) is called the excess. In this paper, we determine the necessary and sufficient conditions for the existence of a simple minimum \( (v, C_5, 1) \)-covering. More precisely, for each \( v \geq 6 \), we prove that there is a simple minimum \( (v, C_5, 1) \)-covering having all possible excesses.