Sigma Chromatic Number of the Middle Graph of Some General Families of Trees of Height 2

Let G be a nontrivial connected graph and letc: V (G) → be a vertex coloring ofG, where adjacent vertices may have the same color. For a vertexv ofG, the color sumσ(v) ofv is the sum of the colors of the vertices adjacent tov. The coloringc is said to be a sigma coloring ofG ifσ(u)σ(v) wheneveru andv are adjacent vertices inG. The minimum number of colors that can be used in a sigma coloring ofG is called the sigma chromatic number ofG and is denoted byσ(G). In this study, we show that the sigma chromatic number of the middle graph of full binary trees of heighth is 2. We also determine a lower bound for the sigma chromatic number of graphs containing an induced subgraph isomorphic to a complete graph joined by pendant vertices. With this lower bound, we obtain the sigma chromatic number of the middle graph of the graphsGn,t,cor(Kn), stars, bistars, and the middle graph of some families of trees of height 2

On the sigma chromatic number of the join of a finite number of paths and cycles

Let G \u3eGG be a simple connected graph and c:V(G)→ℕ \u3ec:V(G)→Nc:V(G)→ℕ a coloring of the vertices in G. \u3eG.G. For any v∈V(G) \u3ev∈V(G)v∈V(G), let σ(v) \u3eσ(v)σ(v) be the sum of colors of the vertices adjacent to v \u3evv. Then c \u3ecc is called a sigma coloring of G \u3eGG if for any two adjacent vertices u,v∈V(G),σ(v)≠σ(u). \u3eu,v∈V(G),σ(v)≠σ(u).u,v∈V(G),σ(v)≠σ(u). The minimum number of colors needed in a sigma coloring of G \u3eGG is the sigma chromatic number of G \u3eGG, denoted by σ(G). \u3eσ(G).σ(G). In this paper; we prescribe a sigma coloring of the join of paths and cycles. As a consequence; we determine the sigma chromatic number of the join of a finite number of paths and cycles. In particular; let G=Σl i=1 Hi where Hi=Pni or Hi= Cni; with 6 ≤ n ≤ 1 ≤ ... ≤ nl. If ni+2 - ni ≥ 2 where 1 ≤ i ≤ l-2 and (H1, H2) ≠ (C6, C6); then σ (G) = 3 if Hi is an odd cycle, for some i, and σ(G) = 2 otherwise

Concerning a Decision-Diagram-Based Solution to the Generalized Directed Rural Postman Problem

Decision-diagram-based solutions for discrete optimization have been persistently studied. Among these is the use of the zero-suppressed binary decision diagram, a compact graph-based representation for a specified family of sets. Such a diagram may work out combinatorial problems by efficient enumeration. In brief, an extension to the frontierbased search approach for zero-suppressed binary decision diagram construction is proposed. The modification allows for the inclusion of a class-determined constraint in formulation. Variations of the generalized directed rural postman problem, proven to be nondeterministic polynomial-time hard, are solved on some rapid transit systems as illustration. Lastly, results are juxtaposed against standard integer programming in establishing the relative superiority of the new technique

Technological Tools for the Teaching and Learning of Statistics

Statistics and its applications form an integral part of STEM education. In the literature, it is shown that technology is valuable in supporting the teaching and learning of statistics. This paper discusses some technological tools that have been developed to support statistics education in the grade school, junior high school, and senior high school levels. It describes the design and pedagogical basis of these tools, and how these may be integrated in the classroom

Design of a Mobile App to Promote Understanding and Fluency in Finding the Equation of a Line

This paper focuses on the design of a mobile app called Pick or Fish that fosters comprehension and mastery of the concepts of slopes, y-intercepts and equations of lines. The app\u27s pedagogical value lies in its potential to help students understand and become proficient in these concepts. The app is suitable for use on low-cost mobile devices. It functions within an engaging game-like setting featuring visual elements that enable students to see the effect of parameter changes on the direction of a line. The beginner and advanced levels of the app have scaffolding features that gradually introduce the students to the key aspects of linear functions. The mechanics of the app, its pedagogical basis and how integration in the classroom may be achieved as teachers plan the lesson, facilitate open-ended discussion and encourage independent use of the app are also discussed

A Mathematical App for the Conceptual Understanding of Area and Perimeter

This paper discusses an app that was developed to build a strong understanding of the concepts of area and perimeter in students. An important feature of the app is the three-component feature which highlights progressive learning: Explore, designed for the learning of the conceptual understanding of area and perimeter; Apply, where area and perimeter concepts are applied; and Create intended for constructing representations to develop higher order thinking skills. The pedagogical basis for the creation of the app, the game design elements employed in the app as well as the integration of the app in the classroom will be presented