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 \(p \equiv 3 \pmod{4}\).

For two vertices \(u\) and \(v\) in a strong oriented graph \(D\), the strong distance \(\operatorname{sd}(u,v)\) between \(u\) and \(v\) is the minimum size (the number of arcs) of a strong sub-digraph of \(D\) containing \(u\) and \(v\). For a vertex \(v\) of \(D\), the strong eccentricity \(\operatorname{se}(v)\) is the strong distance between \(v\) and a vertex farthest from \(v\). The strong radius \(\operatorname{srad}(D)\) is the minimum strong eccentricity among the vertices of \(D\). The strong diameter \(\operatorname{sdiam}(D)\) is the maximum strong eccentricity among the vertices of \(D\). In this paper, we investigate the strong distances in strong oriented complete \(k\)-partite graphs. For any integers \(\delta, r, d\) with \(0 \leq \delta \leq \lceil\frac{k}{2}\rceil, 3 \leq r \leq \lfloor\frac{k}{2}\rfloor, 4 \leq d \leq k\), we have shown that there are strong oriented complete \(k\)-partite graphs \(K’, K”, K”’\) such that \(\operatorname{sdiam}(K’) – \operatorname{srad}(K’) = \delta, \operatorname{srad}(K”) = r\), and \(\operatorname{sdiam}(K”’) = d\).