Let \(k \geq 3\) be an integer, and let \(G\) be a graph of order \(n\) with \(n \geq \max\{10, 4k-3\}\) and \(\delta(G) \geq k+1\). If \(G\) satisfies \(\max\{d_G(x), d_G(y)\} \geq \frac{n}{2}\) for each pair of nonadjacent vertices \(x, y\) of \(G\), then \(G\) is a fractional \(k\)-covered graph. The result is best possible in some sense, and it improves and extends the result of C. Wang and C. Ji (C. Wang and C. Ji, Some new results on \(k\)-covered graphs, Mathematica Applicata \(11(1) (1998), 61-64)\).

For a positive integer \(k\), let \(\mathbb{Z}_k = (\mathbb{Z}_k, +, 0)\) be the additive group of congruences modulo \(k\) with identity \(0\), and \(\mathbb{Z}_1\) is the usual group of integers \(\mathbb{Z}\). We call a finite simple graph \(G = (V(G), E(G))\) \(\mathbb{Z}_k\)-magic if it admits an edge labeling \(\ell: E(G) \to \mathbb{Z}_k \setminus \{0\}\) such that the induced vertex sum labeling \(\ell^+: V(G) \to \mathbb{Z}_k\), defined by \(\ell^+(v) = \sum_{uv \in E(G)} \ell(uv)\), is constant. The constant is called a \emph{magic sum index}, or an \emph{index} for short, of \(G\) under \(\ell\), following R. Stanley. The \emph{null set} of \(G\), defined by E. Salehi as the set of all \(k\) such that \(G\) is \(\mathbb{Z}_k\)-magic with zero magic sum index, is denoted by \(Null(G)\). For a fixed integer \(k\), we consider the set of all possible magic sum indices \(r\) such that \(G\) is \(\mathbb{Z}_k\)-magic with magic sum index \(r\), and denote it by \(I_k(G)\). We call \(I_k(G)\) the \emph{index set} of \(G\) with respect to \(\mathbb{Z}_k\). In this paper, we investigate properties and relations of index sets \(I_k(G)\) and null sets \(Null(G)\) for \(\mathbb{Z}_k\)-magic graphs. Among others, we determine null sets of generalized wheels and generalized fans and construct infinitely many examples of \(\mathbb{Z}_k\)-magic graphs with magic sum zero. Some open problems are presented.

Packing and covering are dual problems in graph theory. A graph \(G\) is called \(H\)-equipackable if every maximal \(H\)-packing in \(G\) is also a maximum \(H\)-packing in \(G\). Dually, a graph \(G\) is called \(H\)-equicoverable if every minimal \(H\)-covering in \(G\) is also a minimum \(H\)-covering in \(G\). In 2012, Zhang characterized two kinds of equipackable paths and cycles: \(P_k\)-equipackable paths and cycles, and \(M_k\)-equipackable paths and cycles. In this paper, we characterize \(P_k\)-equicoverable (\(k > 3\)) paths and cycles, and \(M_k\)-equicoverable (\(k > 2\)) paths and cycles.

For non-negative integers \(n_1, n_2, \ldots, n_t\), let \(GL_{n_1, n_2, \ldots, n_t}(\mathbb{F}_q)\) denote the \(t\)-singular general linear group of degree \(n = n_1 + n_2 + \cdots + n_t\) over the finite field \(\mathbb{F}_q^{n_1+n_2+\ldots+n_t}\) denote the \((n_1+n_2+\ldots+n_t)\)-dimensional \(t\)-singular linear space over the finite \(\mathbb{F}\). Let \(\mathcal{M}\) be any orbit of subspaces under \(GL_{n_1, n_2, \ldots, n_t}(\mathbb{F}_q)\). Denote by \(\mathcal{L}\) the set of all intersections of subspaces in \(M\). Ordered by ordinary or reverse inclusion, two posets are obtained. This paper discusses their geometricity and computes their characteristic polynomials.

The purpose of this paper is to establish g-analogue of some identities and then generalize the result to give identities for finite sums for products of generalized q-harmonic numbers and reciprocals of \(q\)-binomial coefficients.

For a finite group \(G\), let \(P(m,n,G)\) denote the probability that a \(m\)-subset and an \(n\)-subset of \(G\) commute elementwise, and let \(P(n,G) = P(1,n,G)\) be the probability that an element commutes with an \(n\)-subset of \(G\). Some lower and upper bounds are given for \(P(m,n,G)\), and it is shown that \(\{P(m,n,G)\}_{m,n}\) is decreasing with respect to \(m\) and \(n\). Also, \(P(m,n,G)\) is computed for some classes of finite groups, including groups with a central factor of order \(p^2\) and \(P(n,G)\) is computed for groups with a central factor of order \(p^3\) and wreath products of finite abelian groups.

For \(S \subseteq V(G)\) and \(|S| \geq 2\), let \(\lambda(S)\) denote the maximum number of edge-disjoint trees connecting \(S\) in \(G\). For an integer \(k\) with \(2 \leq k \leq n\), the generalized \(k\)-edge-connectivity \(\lambda_k(G)\) of \(G\) is defined as \(\lambda_k(G) = \min\{\lambda(S) : S \subseteq V(G) \text{ and } |S| = k\}\). Note that when \(|S| = 2\), \(\lambda_2(G)\) coincides with the standard \emph{edge-connectivity} \(\lambda(G)\) of \(G\). In this paper, we characterize graphs of order \(n\) such that \(\lambda_n(G) = n – 3\). Furthermore, we determine the minimal number of edges of a graph \(G\) of order \(n\) with \(\lambda_3(G) = 1, n – 3, n – 2\) and establish a sharp lower bound for \(2 \leq \lambda_3(G) \leq n – 4\).

The noncrossing partitions with fixed points have been introduced and studied in the literature. In this paper, as their continuations, we derive expressions for \(f_m(x_1, 0^\mu, x_{\mu+2},0^\rho,x_{\mu+\mu+3},0^{m-\mu-\rho-3})\),and \(f_{m}(x_1,x_2, 0^\mu, x_{\mu+3},0^\rho,x_{\mu+\mu+3},0^{\rho+\mu+4},0^{m-\rho-\mu-4}\), are given,respectively. Moreover, we introduce noncrossing partitions with fixed points having specific property \(\mathcal{P}\) and describe their enumeration through a multivariable function \(f_m^\mathcal{P}(x_1, x_2, \ldots, x_m)\). Additionally, we obtain counting formulas for \(f_m^\mathcal{P}(x_1, 0^{m-1})\) and \(f_m^\mathcal{P}(x_1, x_2, 0^{m-2})\) for various properties \(\mathcal{P}\).

Let \(G = (V(G), E(G))\) be a simple, connected, and undirected graph with vertex set \(V(G)\) and edge set \(E(G)\). A set \(S \subseteq V(G)\) is a \emph{dominating set} if for each \(v \in V(G)\), either \(v \in S\) or \(v\) is adjacent to some \(w \in S\). That is, \(S\) is a dominating set if and only if \(N[S] = V(G)\). The \emph{domination number} \(\gamma(G)\) is the minimum cardinality of minimal dominating sets. In this paper, we provide an improved upper bound on the domination number of generalized Petersen graphs \(P(c,k)\) for \(c \geq 3\) and \(k \geq 3\). We also prove that \(\gamma(P(4k,k)) = 2k + 1\) for even \(k\), \(\gamma(P(5k, k)) = 3k\) for all \(k \geq 1\), and \(\gamma(P(6k,k)) = \left\lceil \frac{10k}{3} \right\rceil\) for \(k \geq 1\) and \(k \neq 2\).

A proper coloring of a graph \(G\) assigns colors to vertices such that adjacent vertices receive distinct colors. The minimum number of colors is the chromatic number \(\chi(G)\). For a graph \(G\) and a proper coloring \(c: V(G) \to \{1, 2, \ldots, k\}\), the color code of a vertex \(v\) is \(code(v) = (c(v), S_v)\), where \(S_v = \{c(u): u \in N(v)\}\). Coloring \(c\) is \emph{singular} if distinct vertices have distinct color codes, and the \emph{singular chromatic number} \(\chi_s(G)\) is the minimum positive integer \(k\) for which \(G\) has a singular \(k\)-coloring. Thus, \(\chi(G) \leq \chi_{si}(G) \leq n\) for every graph \(G\) of order \(n\). We establish a characterization for all triples \((a, b, n)\) of positive integers for which there exists a graph \(G\) of order \(n\) with \(\chi(G) = a\) and \(\chi_{si}(G) = b\). Furthermore, for every vertex \(v\) and edge \(e\) in \(G\), we show:
\( \chi_{si}(G) – 1 \leq \chi_{si}(G – v) \leq \chi_{si}(G) + \deg(v) \) and
\( \chi_{si}(G) – 1 \leq \chi_{si}(G – e) \leq \chi_{si}(G) + 2, \)
and prove that these bounds are sharp. Additionally, we determine the singular chromatic numbers of cycles and paths.