In the realm of graph theory, recent developments have introduced novel concepts, notably the \(\nu\varepsilon\)-degree and \(\varepsilon\nu\)-degree, offering expedited computations compared to traditional degree-based topological indices (TIs). These TIs serve as indispensable molecular descriptors for assessing chemical compound characteristics. This manuscript aims to meticulously compute a spectrum of TIs for silicon carbide \(SiC_{4}\)-\(I[r,s]\), with a specific focus on the \(\varepsilon\nu\)-degree Zagreb index, the \(\nu\varepsilon\)-degree Geometric-Arithmetic index, the \(\varepsilon\nu\)-degree Randić index, the \(\nu\varepsilon\)-degree Atom-bond connectivity index, the \(\nu\varepsilon\)-degree Harmonic index, and the \(\nu\varepsilon\)-degree Sum connectivity index. This study contributes to the ongoing advancement of graph theory applications in chemical compound analysis, elucidating the nuanced structural properties inherent in silicon carbide molecules.

Graph theory has experienced notable growth due to its foundational role in applied mathematics and computer science, influencing fields like combinatorial optimization, biochemistry, physics, electrical engineering (particularly in communication networks and coding theory), and operational research (with scheduling applications). This paper focuses on computing topological properties, especially in molecular structures, with a specific emphasis on the nanotube \(HAC_{5}C_{7}[w,t]\).

Let \(\beta_{H}\) denote the orbit graph of a finite group \(H\). Let \(\zeta\) be the set of commuting elements in \(H\) with order two. An orbit graph is a simple undirected graph where non-central orbits are represented as vertices in \(\zeta\), and two vertices in \(\zeta\) are connected by an edge if they are conjugate. In this article, we explore the Laplacian energy and signless Laplacian energy of orbit graphs associated with dihedral groups of order $2w$ and quaternion groups of order \(2^{w}\).

In this paper, we introduce the concept of the generalized \(3\)-rainbow dominating function of a graph \(G\). This function assigns an arbitrary subset of three colors to each vertex of the graph with the condition that every vertex (including its neighbors) must have access to all three colors within its closed neighborhood. The minimum sum of assigned colors over all vertices of G is defined as the \(g_{3}\)-rainbow domination number, denoted by \(\gamma_{g3r}\). We present a linear-time algorithm to determine a minimum generalized 3-rainbow dominating set for several graph classes: trees, paths \((P_n)\), cycles \((C_n)\), stars (\(K_1,n)\), generalized Petersen graphs \((GP(n,2)\), GP \((n,3))\), and honeycomb networks \((HC(n))\).

A module \(M\) over a commutative ring is termed an \(SCDF\)-module if every Dedekind finite object in \(\sigma[M]\) is finitely cogenerated. Utilizing this concept, we explore several properties and characterize various types of \(SCDF\)-modules. These include local \(SCDF\)-modules, finitely generated $SCDF$-modules, and hollow \(SCDF\)-modules with \(Rad(M) = 0 \neq M\). Additionally, we examine \(QF\) SCDF-modules in the context of duo-ri

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.