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.

We study two-path convexity in bipartite tournaments. For a bipartite tournament, we obtain both a necessary condition and a sufficient condition on the adjacency matrix for its rank to be two. We then investigate 4-cycles in bipartite tournaments of small rank. We show that every vertex in a bipartite tournament of rank two lies on a four cycle, and bipartite tournaments with a maximum number of 4-cycles do not necessarily have minimum rank.

A set \(S\) of vertices in a graph \(G\) is a clique dominating set of \(G\) if \(S\) contains at least one vertex of every clique \(C\) of \(G\). The clique domination number \(\gamma_q(G)\) and the upper clique domination number \(\gamma_q(G)\) are, respectively, the minimum and maximum cardinalities of a minimal clique dominating set of \(G\). In this paper, we prove that the problem of computing \(\Gamma_q(G)\) is NP-complete even for split graphs and the problem of computing \(\gamma_q(G)\) is NP-complete even for chordal graphs. In addition, for a block graph \(BG\) we show that the clique domination number is bounded above by the vertex independence number (\(\gamma_q(BG) \leq \beta(BG)\)) and give a linear algorithm for computing \(\gamma_q(BG)\).