In mathematics education research, mathematics task sets involving mixed practice include tasks from many different topics within the same assignment. In this paper, we use graph decompositions to construct mixed practice task sets for Calculus I, focusing on derivative computation tasks, or tasks of the form “Compute \(f'(x)\) of the function \(f(x)=\) [elementary function].” A decomposition \(D\) of a graph \(G=(V,E)\) is a collection \(\{H_1, H_2, …, H_t\}\) of nonempty subgraphs such that \(H_i=G[E_i]\) for some nonempty subset \(E_i\) of \(E(G)\), and \(\{E_1, E_2, …, E_t\}\) is a partition of \(E(G)\). We extend results on decompositions of the complete directed graph due to Meszka & Skupień to construct balanced task sets that assess the Chain Rule.

Let \(G\) be a graph of order \(n\) and let \(A\) be an additive Abelian group with identity 0. A mapping \(l : V(G) \to A \setminus \{0\}\) is said to be a \(A\)-vertex magic labeling of \(G\) if there exists a \(\mu \in\) \(A\) such that \(w(v) = \sum\limits_{u \in N_G(v)} l(u) = \mu\) for all \(v \in V\) and \(\mu\) is called a magic constant of \(\ell\). The group distance magic set of an \(A\)-vertex magic graph \(gdms(G,A)\) is defined as \(gdms(G,A):= \{ \lambda: \lambda \text{ is a magic constant of some $A$-vertex magic labeling} \}\). In this paper, we investigate under what conditions \(gdms(G,A)\) is a subgroup of \(A\). We also introduce the concept of the reduced group distance magic set, \(rgdms(G, A)\), which can be used as a tool to determine \(gdms(G, A)\).

Let \(2\le k\in\mathbb{Z}\). We say that a total coloring of a \(k\)-regular simple graph via \(k+1\) colors is an efficient total coloring if each color yields an efficient dominating set, where the efficient domination condition applies to the restriction of each color class to the vertex set. We prove that Hamming shells of star transposition graphs and Hamming cubes have efficient total colorings. Also in this work, a focus is set upon the graphs of girth \(2k\) and \(k\). Efficient total colorings of finite connected simple cubic graphs of girth 6 are constructed. These are of two specific types, namely: (a) those whose 6-cycles use just 3 colors with antipodal monochromatic pairs of vertices or edges; (b) those whose 6-cycles do not respect item (a) so they use four colors. An orthogonality property holds for all graphs of type (a). Such property allows further edge-half-girth colorings in the corresponding prism graphs.

A \(\{2\}\)-dominating function (\(\{2 \}\)DF) on a graph \(G=(V(G),E(G))\) is a function \(f : V(G) \rightarrow \{0,1,2 \}\) such that \(f(N[v]) \geq 2\) for every \(v \in V(G)\), where \(N[v]\) is the closed neighourhood of \(v\). The \(\{2\}\)-domination number of \(G\) is the minimum weight \(\omega(f) = \sum\limits_{v \in V(G)} f(v)\) among all \(\{2 \}\)-dominating functions on \(G\). In this article, we prove that if \(G\) and \(H\) are graphs with no isolated vertex, then for any vertex \(v \in V(H)\) there are six closed formulas for the \(\{2\}\)-domination number of the rooted product graph \(G \circ_v H\). We also characterize the graph \(G\) and \(H\) that satisfy each of these formulas.

The chemical graphs are graphs that have no vertex with degree greater than 4. The sigma index of a graph \(G\) is defined by \(\sum_{uv\in E(G)} (deg_{G}(u)-deg_{G}(v))^{2}\), where \(deg_{G}(u)\) stands for the degree of vertex \(u\) in \(G\). In this work, we present lower and upper bounds on the sigma index for chemical trees with a given order and number of pendent vertices. Furthermore, we solve the problem of minimizing sigma index for chemical graphs of order \(n\) having \(m\) edges and \(p\) pendent vertices.

Sports movement recognition is vital for performance assessment, training optimization, and injury prevention, but manual observation is slow and inconsistent. We propose a compact framework that fuses deep learning with biodynamic analysis: convolutional neural networks (CNNs) extract spatial cues from video, a biodynamic encoder derives joint angles, torques, velocities, and forces, and temporal convolutional networks (TCNs) capture sequential dependencies. Using a simulated multimodal dataset of athletic activities, our method outperforms baseline CNN and LSTM models, achieving higher precision (91.5), recall (93.2), and accuracy (92.7). Gains are largest for complex biomechanics (e.g., throwing, kicking), with up to a 10% accuracy increase from biodynamic integration. These results highlight the value of multimodal fusion and provide a scalable path toward real-time, AI-driven sports performance monitoring, with potential extensions to niche sports (fencing, gymnastics, pole vaulting, javelin).

Let \(S\) be an independent set of a connected graph \(G\) of order atleast \(2\). A set \(S' \subseteq V(G)-S\) is an \(S\)-fixed geodetic set of \(G\) if each vertex \(v\) in \(G\) lies on an \(x-y\) geodesic for some \(x\in S\) and \(y\in S'\). The \(S\)-fixed geodetic number \(g_s(G)\) of \(G\) is the minimum cardinality of an \(S\)-fixed geodetic set of \(G\). The independent fixed geodetic number of \(G\) is \(g_{if}(G) = min \left\{g_s(G)\right\}\), where the minimum is taken over all independent sets \(S\) in \(G\). An independent fixed geodetic set of cardinality \(g_{if}(G)\) is called a \(g_{if}\)-set of \(G\). We determine bounds for it and characterize graphs which realize these bounds. Also, the relations with the vertex geodomination number, vertex independence number and vertex covering number of graphs are studied. Some realization results based on the parameter \(g_{if}(G)\) are generated. Finally, two algorithms are designed to compute the independent fixed geodetic number \(g_{if}(G)\) and their complexity results are analyzed.

Enumerative study of RNA secondary structures is one of the most important topics in computational biology. However, most of the existing results are concerned with a single type of structural motifs and are asymptotic. Hairpins and stacks are among the most important motifs in secondary structures. Certain subsets of secondary structures characterized by the number of contained hairpins and the way how these hairpin loops are organized, for instance, cloverleaves (Waterman 1979), have been enumerated in a variety of works, mostly asymptotically. In this paper, we generalize these enumerations and combinatorially obtain exact formulae counting general RNA secondary structures by the joint distribution of hairpins and stacks.

Graph Theory was started by Euler after solving the famous Konigsberg bridge problem. The Graph Coloring is among one of the famous topic for research since it has many beautiful theorems on optimization and its applications in numerous fields of science. The Pi coloring is the coloring of graph parts without a recurring pattern. As a result, it is defined as a function from a set of graph elements with similar properties to the power set of colors, so that each set receives a different color set from the power set. In consequence, Incident Vertex Pi coloring of a graph is defined as the coloring of incident vertices for every single edge with Pi coloring. Incident Vertex Pi coloring of the complete graph is \(n\), wheel graph, star graph and double star graph is \(n+1\), diamond, friendship graphs is \(\Delta +1\), and double fan graph is \(\Delta +2\). In this research, we derived the Incident Vertex PI coloring of Star and Double Star graph’s Middle graph, Total graph, Line graph, and Splitting graph.

For a graph \(G\) with a (not necessarily proper) vertex coloring, a set \(D\subseteq V(G)\) is a polychromatic dominating set of \(G\) if it is a dominating set and each vertex in \(D\) is a different color. The polychromatic domination number of \(G\), \(\rho(G)\), is the minimum number of colors such that, for any \(\rho(G)\)-coloring (with exactly \(\rho(G)\) colors) of the vertices of \(G\), there exists a polychromatic dominating set of \(G\). This paper begins the exploration of the polychromatic domination number. In particular we give tight upper and lower bounds for \(\rho(G)\) both of which are functions of the minimum degree of \(G\).