Rado numbers are closely related to Ramsey numbers, but pertaining to equations and integers instead of cliques within graphs. For every integer \( m \geq 3 \) and every integer \( c \), let the 2-color Rado number \( r(m,c) \) be the least integer, if it exists, such that for every 2-coloring of the set \( \{1,2,\ldots,r(m,c)\} \) there exists a monochromatic solution to the equation \(\sum_{i=1}^{m-1} x_i + c = x_m\) .The values of \( r(m,c) \) have been determined previously for nonnegative values of \( c \), as well as all values of \( m \) and \( c \) such that \( -m+2 < c < 0 \) and \( c < -(m-1)(m-2) \). In this paper, we find \( r(m,c) \) for the remaining values of \( m \) and \( c \).

This paper deals with the Orchard crossing number of some families of graphs which are based on cycles. These include disjoint cycles, cycles which share a vertex and cycles which share an edge. Specifically, we focus on the prism and ladder graphs.

Let \(i(G)\) be the number of isolated vertices in graph \(G\). The isolated toughness of \(G\) is defined as \(I(G) = +\infty\) if \(G\) is complete; \(I(G) = \text{min}\{|S|/i(G-S) : S \subseteq V(G), i(G-S) \geq 2\}\) otherwise. In this paper, we determine that \(G\) is a fractional \((g, f, n)\)-critical graph if \(I(G) \geq \frac{b^2 + bn – 1}{a}\) if \(b > a\); \(I(G) \geq b + n\) if \(a = b\).

Let \(\mathcal{C}\) be a finite family of boxes in \(\mathbb{R}^d\), \(d \geq 3\), with \(S = \cup\{C : C \in \mathcal{C}\}\) connected and \(p \in S\). Assume that, for every geodesic chain \(D\) of \(\mathcal{C}\)-boxes containing \(p\), each coordinate projection \(\pi(D)\) of \(D\) is staircase starshaped with \(\pi(p) \in \text{Ker}\ \pi(D)\). Then \(S\) is staircase starshaped and \(p \in \text{Ker}\ S\). For \(n\) fixed, \(1 \leq n \leq d-2\), an analogous result holds for composites of \(n\) coordinate projections of \(D\) into \((d-n)\)-dimensional flats.

Let \( T(G) \) and \(\text{bind}(G)\) be the tenacity and the binding number, respectively, of a graph \( G \). The inequality \( T(G) \geq \text{bind}(G) – 1 \) was derived by D. Moazzami in [11]. In this paper, we provide a stronger lower bound on \( T(G) \) that is best possible when \(\text{bind}(G) \geq 1\).

In this paper, we give a new look at Sears’ \({}_{3}\phi_{2}\) transformation formula via a discrete random variable. This interpretation may provide a method to calculate \({}_{3}\phi_{2}\) by Monte Carlo experiments.

Symmetry plays a fundamental role in the design of experiments. In particular, symmetries of factorial designs that preserve their statistical properties are exploited to find designs with the best statistical properties. By using a result proved by Rosenberg [1], the concept of the LP relaxation orthogonal array polytope is developed and studied. A complete characterization of the permutation symmetry group of this polytope is made. Also, this characterization is verified computationally for many cases. Finally, a proof is provided.

Let \( R \) be a noncommutative ring with identity and \( Z(R)^* \) be the non-zero zero-divisors of \( R \). The directed zero-divisor graph \(\Gamma(R)\) of \( R \) is a directed graph with vertex set \( Z(R)^* \) and for distinct vertices \( x \) and \( y \) of \( Z(R)^* \), there is a directed edge from \( x \) to \( y \) if and only if \( xy = 0 \) in \( R \). S.P. Redmond has proved that for a finite commutative ring \( R \), if \(\Gamma(R)\) is not a star graph, then the domination number of the zero-divisor graph \(\Gamma(R)\) equals the number of distinct maximal ideals of \( R \). In this paper, we prove that such a result is true for the noncommutative ring \( M_2(\mathbb{F}) \), where \(\mathbb{F}\) is a finite field. Using this, we obtain a class of graphs for which all six fundamental domination parameters are equal.

Multilevel Hadamard matrices (MHMs), whose entries are integers as opposed to the traditional restriction to \(\{\pm 1\}\), have been introduced as a way to construct multilevel zero-correlation zone sequences for use in approximately synchronized code division multiple access (AS-CDMA) systems. This paper provides a construction technique to produce \(2^m \times 2^m\) MHMs whose \(2^m\) alphabet entries form an arithmetic progression, up to sign. This construction improves upon existing constructions because it permits control over the spacing and overall span of the MHM entries. MHMs with such regular alphabets are a more direct generalization of traditional Hadamard matrices and are thus expected to be more useful in applications analogous to those of Hadamard matrices. This paper also introduces mixed-circulant MHMs which provide a certain advantage over known circulant MHMs of the same size.

MHMs over the Gaussian (complex) and Hamiltonian (quaternion) integers are introduced. Several constructions are provided, including a generalization of the arithmetic progression construction for MHMs over real integers. Other constructions utilize amicable pairs of MHMs and c-MHMs, which are introduced as natural generalizations of amicable orthogonal designs and c-Hadamard matrices, respectively. The constructions are evaluated against proposed criteria for interesting and useful MHMs over these generalized alphabets.

A family \(\mathcal{G}\) of connected graphs is a family with constant metric dimension if \(\text{dim}(G)\) is finite and does not depend upon the choice of \(G\) in \(\mathcal{G}\). In this paper, we show that the sunlet graphs, the rising sun graphs, and the co-rising sun graphs have constant metric dimension.