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.

By means of inversion techniques, new proofs for Whipple’s transformation and Watson’s \(q\)-Whipple transformation are offered.

In this paper, we introduced the notion of left-right and right-left \(f\)-derivations of a \(B\)-algebra and investigated some related properties. We studied the notion of \(f\)-derivation of a \(0\)-commutative \(B\)-algebra and stated some related properties.

Let \(G\) be a \(k\)-edge connected simple graph with \(k \leq 3\), minimal degree \(\delta(G) \geq 3\), and girth \(g\), where \(r = \left\lfloor \frac{g-1}{2} \right\rfloor\). If the independence number \(\alpha(G)\) of \(G\) satisfies

\[\alpha(G) < \frac{6{(\delta-1)}^{\lfloor\frac{g}{2}\rfloor}-6}{(4-k)(\delta-2)} – \frac{6(g-2r-1)}{4-k} \] then \(G\) is up-embeddable.

Let \(p\) be a prime number such that \(p \equiv 1, 3 \pmod{4}\), let \(\mathbb{F}_p\) be a finite field, and let \(N \in \mathbb{F}_p^* = \mathbb{F}_p – \{0\}\) be a fixed element. Let \(P_p^k(N): x^2 – ky^2 = N\) and \(\tilde{P}_p^k(N): x^2 + 2y – ky^2 = N\) be two Pell equations over \(\mathbb{F}_p\), where \(k = \frac{p-1}{4}\) or \(k = \frac{p-3}{4}\), respectively. Let \(P_p^k(N)(\mathbb{F}_p)\) and \(\tilde{P}_p^k(N)(\mathbb{F}_p)\) denote the set of integer solutions of the Pell equations \(P_p^k(N)\) and \(\tilde{P}_p^k(N)\), respectively. In the first section, we give some preliminaries from the general Pell equation \(x^2 – ky^2 = \pm N\). In the second section, we determine the number of integer solutions of \(P_p^k(N)\). We prove that \(P_p^k(N)(\mathbb{F}_p) = p+1\) if \(p \equiv 1 \pmod{4}\) or \(p \equiv 7 \pmod{12}\) and \(P_p^k(N)(\mathbb{F}_p) = p-1\) if \(p \equiv 11 \pmod{12}\). In the third section, we consider the Pell equation \(\tilde{P}_p^k(N)\). We prove that \(\tilde{P}_p^k(N)(\mathbb{F}_p) = 2p\) if \(p \equiv 1 \pmod{4}\) and \(N \in Q_p\); \(\tilde{P}_p^k(N)(\mathbb{F}_p) = 0\) if \(p \equiv 1 \pmod{4}\) and \(N \notin Q_p\); \(\tilde{P}_p^k(N)(\mathbb{F}_p) = p+1\) if \(p \equiv 3 \pmod{4}\).