Let an \(H\)-point be a vertex of a tiling of \(\mathbb{R}^2\) by regular hexagons of side length 1, and \(D(n)\) a circle of radius \(n\) (\(n \in \mathbb{Z}^+\)) centered at an \(H\)-point. In this paper, we present an algorithm to calculate the number, \(\mathcal{N}_H(D(n))\), of H-points that lie inside or on the boundary of \(D(n)\). Furthermore, we show that the ratio \(\mathcal{N}_H(D(n))/n^2\) tends to \(\frac{2\pi}{S}\) as \(n\) tends to \(\infty\), where \(S = \frac{3\sqrt{3}}{2}\) is the area of the regular hexagonal tiles.

Let \(G\) be a finite, simple graph. We denote by \(\gamma(G)\) the domination number of \(G\). The bondage number of \(G\), denoted by \(b(G)\), is the minimum number of edges of \(G\) whose removal increases the domination number of \(G\). \(C_n\) denotes the cycle of \(n\) vertices. For \(n \geq 5\) and \(n \neq 5k + 3\), the domination number of \(C_5 \times C_n\) was determined in [6]. In this paper, we calculate the domination number of \(C_5 \times C_n\) for \(n = 5k + 3\) (\(k \geq 1\)), and also study the bondage number of this graph, where \(C_5 \times C_n\) is the cartesian product of \(C_5\) and \(C_n\).

A vertex cut that separates the connected graph into components such that every vertex in these components has at least \(g\) neighbors is an \(R^g\)-vertex-cut. \(R^g\)-vertex-connectivity, denoted by \(\kappa^g(G)\), is the cardinality of a minimum \(R^g\)-vertex-cut of \(G\). In this paper, we will determine \(\kappa^g\) and characterize the \(R^g\)-vertex-atom-part for the first and second type Harary graphs.

A graph \(G\) is supereulerian if \(G\) has a spanning eulerian subgraph. We use \(\mathcal{SL}\) to denote the families of supereulerian graphs. In 1995, Zhi-Hong Chen and Hong-Jian Lai presented the following open problem [2, problem 8.8]: Determine

\[L=\min\max\limits_{G\in SL-\{K_1\}}\{\frac{|E(H)|}{|E(G)|} : H \text{ is spanning eulerian subgroup of G}\}.\]

For a graph \(G\), \(O(G)\) denotes the set of all odd-degree vertices of \(G\). Let \(G\) be a simple graph and \(|O(G)| = 2k\). In this note, we show that if \(G\in{SL}\) and \(k \leq 2\), then \(L \geq \frac{2}{3}\).

It is known that the number of Dyck paths is given by a Catalan number. Dyck paths are represented as plane lattice paths which start at the origin \(O\) and end at the point \(P_n = (n,n)\) repeating \((1,0)\) or \((0,1)\) steps without going above the diagonal line \(OP_n\). Therefore, it is reasonable to ask of any positive integers \(a\) and \(b\) what number of lattice paths start at \(O\) and end at point \(A = (a, b)\) repeating the same steps without going above the diagonal line \(OA\). In this article, we show a formula to represent the number of such generalized Dyck paths.

Let \(G\) be a graph with vertex set \(V(G)\) and edge set \(E(G)\), and let \(A\) be an abelian group. A labeling \(f : V(G) \to A\) induces an edge labeling \(f^* : E(G) \to A\) defined by \(f^*(xy) = f(x) + f(y)\), for each edge \(xy \in E(G)\). For \(i \in A\), let \(v_f(i) = \mathrm{card}\{v \in V(G) : f(v) = i\}\) and \(e_f(i) = \mathrm{card}\{e \in E(G) : f^*(e) = i\}\). Let \(c(f) = \{|e_f(i) – e_f(j)|: (i, j) \in A \times A\}\). A labeling \(f\) of a graph \(G\) is said to be \(A\)-friendly if \(|v_f(i)- v_f(j)| \leq 1\) for all \((i, j) \in A \times A\). If \(c(f)\) is a \((0, 1)\)-matrix for an \(A\)-friendly labeling \(f\), then \(f\) is said to be \(A\)-cordial. When \(A = \mathbb{Z}_2\), the friendly index set of the graph \(G\), \(FI(G)\), is defined as \(\{|e_f(0) – e_f(1)| : \text{the vertex labeling } f \text{ is } \mathbb{Z}_2\text{-friendly}\}\). In [13] the friendly index set of cycles are completely determined. In this paper we describe the friendly index sets of cycles with parallel chords. We show that for a cycle with an arbitrary non-empty set of parallel chords, the numbers in its friendly index set form an arithmetic progression with common difference 2.

The eccentricity \(e(v)\) of a vertex \(v\) in a connected graph \(G\) is the distance between \(v\) and a vertex furthest from \(v\). The center \(C(G)\) is the subgraph induced by those vertices whose eccentricity is the radius of \(G\), denoted \(\mathrm{rad}G\), and the periphery \(P(G)\) is the subgraph induced by those vertices with eccentricity equal to the diameter of \(G\), denoted \(\mathrm{diam}G\). The annulus \(\mathrm{Ann}(G)\) is the subgraph induced by those vertices with eccentricities strictly between the radius and diameter of \(G\). In a graph \(G\) where \(\mathrm{rad}G < \mathrm{diam}G\), the interior of \(G\) is the subgraph \(\mathrm{Int}(G)\) induced by the vertices \(v\) with \(e(v) < \mathrm{diam}G\). Otherwise, if \(\mathrm{rad}G = \mathrm{diam}G\), then \(\mathrm{Int}(G) = G\). Another subgraph for a connected graph \(G\) with \(\mathrm{rad}G < \mathrm{diam}G\), called the exterior of \(G\), is defined as the subgraph \(\mathrm{Ext}(G)\) induced by the vertices \(v\) with \(\mathrm{rad}G < e(v)\). As with the interior, if \(\mathrm{rad}G = \mathrm{diam}G\), then \(\mathrm{Ext}(G) = G\). In this paper, the annulus, interior, and exterior subgraphs in trees are characterized.

This paper investigates the dihedral group as the array stabilizer of an augmented \(k\)-set of mutually orthogonal Latin squares. Necessary conditions for the stabilizer to be a dihedral group are established. A set of two-variable identities essential for a dihedral group to be contained in an array stabilizer are determined. Infinite classes of models that satisfy the identities are constructed.

A proper vertex coloring of a graph \(G = (V, E)\) is acyclic if \(G\) contains no bicolored cycle. A graph \(G\) is acyclically \(L\)-list colorable if for a given list assignment \(L = \{L(v) : v \in V\}\), there exists a proper acyclic coloring \(\phi\) of \(G\) such that \(\phi(v) \in L(v)\) for all \(v \in V(G)\). If \(G\) is acyclically \(L\)-list colorable for any list assignment with \(|L(v)| = k\) for all \(v \in V\), then \(G\) is acyclically \(k\)-choosable. In this paper, it is proved that every toroidal graph without 4- and 6-cycles is acyclically \(5\)-choosable.

The centro-polyhedral group \(\langle l,m,n\rangle\), for \(l, m, n \in \mathbb{Z}\), is defined by the presentation
\[\langle x, y, z : x^l = y^m = z^n = xyz \rangle.\]
In this paper, we obtain the periods of \(k\)-nacci sequences in centro-polyhedral groups and related groups.