We study the area distribution of closed walks of length \( n \), starting and ending at the origin. The concept of algebraic area of a walk in the square lattice is slightly modified and the usefulness of this concept is demonstrated through a simple argument. The idea of using a generating function of the form \( (x + x^{-1} + y + y^{-1})^n \) to study these walks is then discussed from a special viewpoint. Based on this, a polynomial time algorithm for calculating the exact distribution of such walks for a given length is concluded. The presented algorithm takes advantage of the Chinese remainder theorem to overcome the problem of arithmetic with large integers. Finally, the results of the implementation are given for \( n = 32, 64, 128 \).

The Wiener polarity index of a graph \( G \) is the number of unordered pairs of vertices \( u, v \) such that the distance between \( u \) and \( v \) is three, which was introduced by Harold Wiener in 1947. A linear time algorithm for computing the Wiener polarity index of trees was described, and also an algorithm which computes the index \( W_p(G) \) for any given connected graph \( G \) on \( n \) vertices in time \( O(M(n)) \) was presented, where \( M(n) \) denotes the time necessary to multiply two \( n \times n \) matrices of small integers (which is currently known to be \( O(n^{2.376}) \)). In this paper, we establish one polynomial algorithm to calculate the value of the Wiener polarity index of a bipartite graph.

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.