We prove a two-point concentration for the tree domination number of the random graph \(G_{n,p}\) provided \(p\) is constant or \(p \to 0\) sufficiently slow.

A 2-independent set in a graph \(G\) is a subset \(J\) of the vertices such that the distance between any two vertices of \(J\) in \(G\) is at least three. The number of 2-independent sets of a graph \(G\) is denoted by \(i_2(G)\). For a forest \(F\), \(i_2(F – e) > i_2(F)\) for each edge \(e\) of \(F\). Hence, we exclude all forests having isolated vertices. A forest is said to be extra-free if it has no isolated vertex. In this paper, we determine the \(k\)-th largest number of 2-independent sets among all extra-free forests of order \(n \geq 2\), where \(k = 1, 2, 3\). Extremal graphs achieving these values are also given.

The notion of multiparameter \(q\)-noncentral Stirling numbers is introduced by means of a triangular recurrence relation. Some properties for these \(q\)-analogues such as vertical and horizontal recurrence relations, horizontal generating functions, explicit formula, orthogonality and inverse relations are established. Moreover, we introduce the multiparameter Bell numbers and Bell polynomials, their connection to factorial moments and their respective \(q\)-analogues.

Let \(a, b\), and \(k\) be nonnegative integers with \(2 \leq a \leq 6\) and \(b \equiv 0 \pmod{a-1}\). Let \(G\) be a graph of order \(n\) with \(n \geq \frac{(a+b-1)(2a+b-4)-a+1}{b} + k\). A graph \(G\) is called an \((a, b, k)\)-critical graph if after deleting any \(k\) vertices of \(G\), the remaining graph has an \([a, b]\)-factor. In this paper, it is proved that \(G\) is an \((a, b, k)\)-critical graph if and only if \[|N_G(X)| >\frac{(a-1)n + |X| + bk-1}{a+b-1} \] for every non-empty independent subset \(X\) of \(V(G)\), and \[\delta(G) > \frac{(a-1)n + b + bk}{a+b-1}.\] Furthermore, it is shown that the result in this paper is best possible in some sense.

Two-dimensional codes in \(LRTJ\) spaces are subspaces of the space \(Mat_{m\times s}(\mathbb{Z}_q)\), the linear space of all \(m \times s\)-matrices with entries from a finite ring \(\mathbb{Z}_q\), endowed with the \(LRTJ\)-metric \([3,9]\). Also, the error-correcting capability of a linear code depends upon the number of parity-check symbols. In this paper, we obtain a lower bound on the number of parity checks of two-dimensional codes in \(LRTJ\)-spaces correcting both independent as well as cluster array errors.

Let \(G = (V, E)\) be a graph without an isolated vertex. A set \(D \subseteq V(G)\) is a total dominating set if \(D\) is dominating and the induced subgraph \(G[D]\) does not contain an isolated vertex. The total domination number of \(G\) is the minimum cardinality of a total dominating set of \(G\). A set \(D \subseteq V(G)\) is a total outer-connected dominating set if \(D\) is total dominating and the induced subgraph \(G[V(G) – D]\) is connected. The total outer-connected domination number of \(G\) is the minimum cardinality of a total outer-connected dominating set of \(G\). We characterize all unicyclic graphs with equal total domination and total outer-connected domination numbers.

We give a characterization of strongly multiplicative graphs. First, we introduce some necessary conditions for a graph to be strongly multiplicative.Second, we discuss the independence of these necessary conditions. Third, we show that they are altogether not sufficient for a graph to be strongly multiplicative. Fourth, we add another necessary condition. Fifth, we prove that this necessary condition is stronger than the mentioned necessary conditions except one. Finally, we conjecture that the condition itself is stronger than all of them.

Let \(G = (V, E)\) be a simple connected graph with \(n\) vertices and \(m\) edges. Further, let \(\lambda_i(L)\), \(i = 1, 2, \ldots, n\), be the non-increasing eigenvalues of the normalized Laplacian matrix of the graph \(G\). In this paper, we obtain the following result: For a connected graph \(G\) of order \(n\), \(lambda_2(L) = \lambda_3(L) = \cdots = \lambda_{n-1}(L)\) if and only if \(G\) is a complete graph \(K_n\) or \(G\) is a complete bipartite graph \(K_{p,q}\). Moreover, we present lower and upper bounds for the normalized Laplacian spectral radius of a graph and characterize graphs for which the lower or upper bounds are attained.

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.