Euler and Lagrange: Graph Theory and Physics Applications

Pronunciation and Etymology

• Euler: Pronounced exactly like the word "oil," hence "Oil-er." It is not pronounced "Yool-er."

• Lagrange: Pronounced in the French style ("La-grange"), not rhyming with the English word "strange."

Introduction to Graph Theory and Leonhard Euler

• The Seven Bridges of Königsberg Problem: Leonhard Euler was originally presented with a logistical problem involving bridges. He found that looking at the actual city map was too complex because of the various roads, buildings, and geographic features.

• Simplification of Maps: To solve this, Euler defined a new type of visualization. Instead of a geographic map, he simplified the problem into a collection of dots and lines. This was the birth of what is now called a graph.

• Comparison to Cartesian Graphs: Unlike the standard Cartesian graphs used in algebra (which feature an $x$-axis and a $y$-axis), Euler’s graphs are abstract representations focused on connectivity rather than coordinates.

• Definition of a Graph: A graph is defined as a collection of dots, which Euler called vertices, connected to one another by lines, which he called edges.

• Discrete Mathematics: This study of graphs falls under the branch of mathematics known as discrete mathematics, an area Euler explored extensively.

Vertices, Edges, and Degrees (Analytical Example: Figure A)

In a hypothetical graph called Figure A, constructed to illustrate Euler's concepts, the following components are identified:

• Vertices (Nodes): These are the individual points or dots in the graph. In Figure A, the vertices are explicitly named:

  • $a, b, c, d, e, f$

• Edges: These are the lines that connect two vertices directly. There must be no other vertex in between the two endpoints for the line to be considered a single edge. The edges in Figure A include:

  • $ab$

  • $ae$

  • $bc$

  • $bd$

  • $ce$

  • $de$

  • $ef$

  • Total Count: There are seven total edges in this specific graph. Note that a connection like $ac$ is not considered an edge if there is no direct line without an intermediate point.

• Degrees: The degree of a vertex represents the number of edges that connect to that specific point. Calculating the degrees for Figure A:

  • $\text{Degree of } a = 2$ (Edges connecting are $ab$ and $ae$)

  • $\text{Degree of } b = 3$

  • $\text{Degree of } c = 2$

  • $\text{Degree of } d = 2$

  • $\text{Degree of } e = 4$ (The highest degree in this graph, with 4 connecting lines)

  • $\text{Degree of } f = 1$ (The lowest degree in this graph)

Eccentricity, Radius, and Diameter

Eccentricity is a measure used to describe the distance between points in a graph, specifically within linear or connected structures.

• Definition of Eccentricity: The distance from a specific vertex to the vertex that is farthest away from it.

• Linear Example (Points a through g): Consider a line of vertices labeled $a, b, c, d, e, f, g$, spaced evenly.

  • Eccentricity of $c$: The distance from $c$ to $a$ is 2. The distance from $c$ to $g$ is 4. Since 4 is the farther distance, the eccentricity of $c$ is $4$.

  • Eccentricity of $f$: The distance from $f$ to $g$ is 1. The distance from $f$ to $a$ is 5. Therefore, the eccentricity of $f$ is $5$.

  • Eccentricity of $d$ (The Middle Point): The distance to $a$ is 3, and the distance to $g$ is 3. Since the furthest distance is 3, the eccentricity of $d$ is $3$.

• The Radius: Defined as the smallest possible eccentricity in a graph. In the example above, the point $d$ provides the radius because its eccentricity ($3$) is lower than any other point. It typically represents the "center" of the structure.

• The Diameter: Defined as the largest possible eccentricity in a graph. In the example above, the diameter is $6$ (the distance from one end of the graph, point $a$, to the other end, point $g$).

Lagrange’s Physics Formula: Velocity and Height

Joseph-Louis Lagrange identified a specific mathematical relationship in physics and calculus regarding the velocity of an object hitting the ground and the height from which it was dropped.

• The Formula:     $V = 8 \times \sqrt{h}$

  • $V$ = Velocity (typically measured in $\text{feet per second}$, unless otherwise specified).

  • $h$ = Height (measured in $\text{feet}$).

• Example 1: Calculating Velocity from a Known Height

  • Problem: If an object is dropped from a height of $100\,\text{feet}$, at what velocity will it hit the ground?

  • Step 1: Plug the height into the formula structure: $V = 8 \times \sqrt{100}$.

  • Step 2: Solve the square root: $V = 8 \times 10$.

  • Step 3: Final calculation: $V = 80\,\text{feet per second}$.

• Example 2: Calculating Height from a Known Velocity

  • Problem: If an object hits the ground at a velocity of $40\,\text{feet per second}$, from what height was it dropped?

  • Step 1: Plug the velocity into the formula: $40 = 8 \times \sqrt{h}$.

  • Step 2: Isolate the square root by dividing both sides by 8: $5 = \sqrt{h}$.

  • Step 3: Eliminate the square root by squaring both sides: $5^2 = h$.

  • Step 4: Final calculation: $h = 25\,\text{feet}$.

Homework and Textbook Errata

• Extra Problem Recommendation: Students are encouraged to practice finding the radius and diameter of a line of vertices from $a$ through $i$. This specific problem type is highly likely to appear on future homework or quizzes.

• Textbook Warning: Note that the answer provided in the back of the textbook for Problem #14 is incorrect. Students should rely on the methodology taught in class rather than checking against the book's key for that specific item.

• Real-World Application: Future worksheets will apply Euler's graph concepts to real-life networks, such as airline routes and logistics networks.