Let \(G=(V(G),E(G))\) be a graph with \(p\) vertices and \(q\) edges. A graph \(G\) of size \(q\) is said to be odd graceful if there exists an injection \(\lambda: V(G) \to {0,1,2,\ldots,2q-1}\) such that assigning each edge \(xy\) the label or weight \(|\lambda(x) – \lambda(y)|\) results in the set of edge labels being \({1,3,5,\ldots,2q-1}\). This concept was introduced in 1991 by Gananajothi. In this paper, we examine the odd graceful labeling of the \(W\)-tree, denoted as \(WT(n,k)\).

This paper introduces a novel type of convex function known as the refined modified \((h,m)\)-convex function, which is a generalization of the traditional \((h,m)\)-convex function. We establish Hadamard-type inequalities for this new definition by utilizing the Caputo \(k\)-fractional derivative. Specifically, we derive two integral identities that involve the nth order derivatives of given functions and use them to prove the estimation of Hadamard-type inequalities for the Caputo \(k\)-fractional derivatives of refined modified \((h,m)\)-convex functions. The results obtained in this research demonstrate the versatility of the refined modified \((h,m)\)-convex function and the usefulness of Caputo \(k\)-fractional derivatives in establishing important inequalities. Our work contributes to the existing body of knowledge on convex functions and offers insights into the applications of fractional calculus in mathematical analysis. The research findings have the potential to pave the way for future studies in the area of convex functions and fractional calculus, as well as in other areas of mathematical research.

In this paper, we utilize the \(\sigma\) category to introduce \(EKFN\)-modules, which extend the concept of the \(EKFN\)-ring. After presenting some properties, we demonstrate, under certain hypotheses, that if \(M\) is an \(EKFN\)-module, then the following equivalences hold: the class of uniserial modules coincides with the class of \(cu\)-uniserial modules; \(EKFN\)-modules correspond to the class of locally noetherian modules; and the class of \(CD\)-modules is a subset of the \(EKFN\)-modules.

Let \(G\) be a finite solvable group and \(\Delta\) be the subset of \(\Upsilon \times \Upsilon\), where \(\Upsilon\) is the set of all pairs of size two commuting elements in \(G\). If \(G\) operates on a transitive \(G\) – space by the action \((\upsilon_{1},\upsilon_{2})^{g}=(\upsilon_{1}^{g},\upsilon_{2}^{g})\); \(\upsilon_{1},\upsilon_{2} \in \Upsilon\) and \(g \in G\), then orbits of \(G\) are called orbitals. The subset \(\Delta_{o}=\{(\upsilon,\upsilon);\upsilon \in \Upsilon, (\upsilon,\upsilon) \in \Upsilon \times \Upsilon\}\) represents \(G’s\) diagonal orbital.
The orbital regular graph is a graph on which \(G\) acts regularly on the vertices and the edge set. In this paper, we obtain the orbital regular graphs for some finite solvable groups using a regular action. Furthermore, the number of edges for each of a group’s orbitals is obtained.

By combining the telescoping method with Cassini–like formulae, we evaluate, in closed forms, four classes of sums about products of two arctangent functions with their argument involving Pell and Pell–Lucas polynomials. Several infinite series identities for Fibonacci and Lucas numbers are deduced as consequences.