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 \(\operatorname{sdiam}(K”’) = d\).

The \(t\)-pebbling number \(f_t(G)\) of a graph \(G\) is the least positive integer \(m\) such that however these \(m\) pebbles are placed on the vertices of \(G\), we can move \(t\) pebbles to any vertex by a sequence of moves, each move taking two pebbles off one vertex and placing one on an adjacent vertex. In this paper, we study the generalized Graham’s pebbling conjecture \(f_t(G \times H) \leq f(G)f_t(H)\) for the product of graphs when \(G\) is a complete \(r\)-partite graph and \(H\) has a \(2t\)-pebbling property.

The detour index of a connected graph is defined as the sum of detour distances between all its unordered vertex pairs. We determine the maximum detour index of \(n\)-vertex unicyclic graphs with maximum degree \(\Delta\), and characterize the unique extremal graph, where \(2 \leq \Delta \leq {n-1}\).

In this study, we obtain the relations among \(k\)-Fibonacci, \(k\)-Lucas, and generalized \(k\)-Fibonacci numbers. Then, we define circulant matrices involving \(k\)-Lucas and generalized \(k\)-Fibonacci numbers. Finally, we investigate the upper and lower bounds for the norms of these matrices.

Let \(G = (V(G), E(G))\) be a graph. A set \(S \subseteq V(G)\) is a dominating set if every vertex of \(V(G) – S\) is adjacent to some vertices in \(S\). The domination number \(\gamma(G)\) of \(G\) is the minimum cardinality of a dominating set of \(G\). In this paper, we study the domination number of the circulant graphs \(C(n; \{1, 2\})\), \(C(n; \{1, 3\})\), and \(C(n; \{1, 4\})\) and determine their exact values.

The Merrifield-Simmons index of a graph \(G\), denoted by \(i(G)\), is defined to be the total number of its independent sets, including the empty set. Let \(\theta(a_1, a_2, \ldots, a_k)\) denote the graph obtained by connecting two distinct vertices with \(k\) independent paths of lengths \(a_1, a_2, \ldots, a_k\) respectively, we named it as multi-bridge graphs for convenience. Tight upper and lower bounds for the Merrifield-Simmons index of \(\theta(a_1, a_2, \ldots, a_k)\) are established in this paper.