Let \(G\) be a graph of order \(n\), and let \(a, b, k\) be nonnegative integers with \(1 \leq a \leq b\). A spanning subgraph \(F\) of \(G\) is called an \([a, b]\)-factor if \(a \leq d_F(x) \leq b\) for each \(x \in V(G)\). Then a graph \(G\) is called an \((a, b, k)\)-critical graph if \(G – N\) has an \([a, b]\)-factor for each \(N \subseteq V(G)\) with \(|N| = k\). In this paper, it is proved that \(G\) is an \((a, b, k)\)-critical graph if \(n \geq \frac{(a+b-1)(a+b-2)}{b} +\frac{bk}{b-1}\), \(bind(G) \geq \frac{(a+b-1)(n-1)}{b(n-1-k)}\), and \(\delta(G) \neq \left\lfloor \frac{(a-1)n+a+b+bk-2}{a+b-1} \right\rfloor\).

The vulnerability shows the resistance of the network until communication breakdown after the disruption of certain stations or communication links. This study introduces a new vulnerability parameter, neighbor rupture degree. The neighbor rupture degree of a non-complete connected graph \(G\) is defined to be

\[Nr(G) = \max\{w(G/S) – |S| – c(G/S): S \subset V(G), w(G/S) \geq 1\}\]

where \(S\) is any vertex subversion strategy of \(G\), \(w(G/S)\) is the number of connected components in \(G/S\), and \(c(G/S)\) is the maximum order of the components of \(G/S\). In this paper, the neighbor rupture degree of some classes of graphs are obtained and the relations between neighbor rupture degree and other parameters are determined.

A set \(S\) of vertices of a graph \(G = (V, E)\) without isolated vertices is a total dominating set if every vertex of \(V(G)\) is adjacent to some vertex in \(S\). The total domination number \(\gamma_t(G)\) is the minimum cardinality of a total dominating set of \(G\). The total domination subdivision number \(sd_{\gamma t}(G)\) is the minimum number of edges that must be subdivided (each edge in \(G\) can be subdivided at most once) in order to increase the total domination number. In this paper, we first prove that \(sd_{\gamma t}(G) \leq n – \delta + 2\) for every simple connected graph \(G\) of order \(n \geq 3\). We also classify all simple connected graphs \(G\) with \(sd_{\gamma t}(G) = n – \delta + 2, n – \delta + 1\), and \(n – \delta\).

In this paper, we obtain the following upper and lower bounds for \(q\)-factorial \([n]_q!\):

\[(q; q)_\infty (1 – q)^{-n} e^{f_q(n+1)} < [n]_q! < (q; q)_\infty (1 – q)^{-n} e^{g_q(n+1)},\] where \(n \geq 1\), \(0 < q < 1\), and the two sequences \(f_q(n)\) and \(g_q(n)\) tend to zero through positive values. Also, we present two examples of the two sequences \(f_q(n)\) and \(g_q(n)\).

A set \(S\) of vertices of a graph \(G = (V, E)\) without isolated vertices is a total dominating set if every vertex of \(V(G)\) is adjacent to some vertex in \(S\). The total domination number \(\gamma_t(G)\) is the minimum cardinality of a total dominating set of \(G\). The total domination subdivision number \(sd_{\gamma t}(G)\) is the minimum number of edges that must be subdivided (each edge in \(G\) can be subdivided at most once) in order to increase the total domination number. In this paper, we first prove that \(sd_{\gamma t}(G) \leq n – \delta + 2\) for every simple connected graph \(G\) of order \(n \geq 3\). We also classify all simple connected graphs \(G\) with \(sd_{\gamma t}(G) = n – \delta + 2, n – \delta + 1\), and \(n – \delta\).

In this paper, we show that the sequences \(p(n, k) := 2^{n-2k} \binom{n-k}{k}\) and \(q(n,k) := 2^{n-2k}\frac{n}{n-k}\binom{n-k}{k}\), \(k = 0, \ldots, \lfloor \frac{n}{2} \rfloor\), are strictly log-concave and then unimodal with at most two consecutive modes. We localize the modes and the integers where there is a plateau. We also give a combinatorial interpretation of \(p(n, k)\) and \(q(n, k)\). These sequences are associated respectively to the Pell numbers and the Pell-Lucas numbers, for which we give some trigonometric relations.

For a finite field \(\mathbb{F}_{p^t}\) of order \(p^t\), where \(p\) is a prime and \(t \geq 1\), we consider the digraph \(G(\mathbb{F}_{p^t}, k)\) that has all the elements of \(\mathbb{F}_{p^t}\) as vertices and a directed edge \(E(a, b)\) if and only if \(a^k = b\), where \(a, b \in \mathbb{F}_{p^t}\). We completely determine the structure of \(G(\mathbb{F}_{p^t},k)\), the isomorphic digraphs of \(\mathbb{F}_{p^t}\), and the longest cycle in \(G(\mathbb{F}_{p^t}, k)\).

Let \(c(H)\) denote the number of components of a graph \(H\). Win proved in \(1989\) that if a connected graph \(G\) satisfies
\[c(G \setminus S) \leq (k – 2)|S| + 2,\text{for every subset S of V(G)},\]
then \(G\) has a spanning tree with maximum degree at most \(k\).

For a spanning tree \(T\) of a connected graph, the \(k\)-excess of a vertex \(v\) is defined to be \(\max\{0, deg_T(v) – k\}\). The total \(k\)-excess \(te(T, k)\) is the summation of the \(k\)-excesses of all vertices, namely,
\[te(T, k) = \sum_{v \in V(T)} \max\{0, deg_T(v) – k\}.\]
This paper gives a sufficient condition for a graph to have a spanning tree with bounded total \(k\)-excess. Our main result is as follows.

Suppose \(k \geq 2\), \(b \geq 0\), and \(G\) is a connected graph satisfying the following condition:
\[\text{for every subset S of V(G)}, \quad c(G \setminus S) \leq (k – 2)|S| + 2+b.\]
Then, \(G\) has a spanning tree with total \(k\)-excess at most \(b\).

A connected graph \(G\) is called \(l_1\)-embeddable, if \(G\) can be isometrically embedded into the \(i\)-space. The hexagonal Möbius graphs \(H_{2m,2k}\) and \(H_{2m+1,2k+1}\) are two classes of hexagonal tilings of a Möbius strip. The regular quadrilateral Möbius graph \(Q_{p,q}\) is a quadrilateral tiling of a Möbius strip. In this note, we show that among these three classes of graphs only \(H_{2,2}\), \(H_{3,3}\), and \(Q_{2,2}\) are \(l_1\)-embeddable.

The boundedness and compactness of the generalized composition operator from \(\mu\)-Bloch spaces to mixed norm spaces are completely characterized in this paper.