Mathematics of Graphs and Euler Circuits

A set of points called vertices and line segments or curves called edges.

Two vertices are considered adjacent if they share a common edge.

Two edges are considered adjacent if they share a common vertex.

An edge where both endpoints are the same vertex.

Two or more edges that connect the same two vertices.

A graph where there is no path to connect any two vertices.

A graph where any two vertices can be connected by a path.

A graph with no edges. It's like a blank canvas. Each vertex stands alone.

A graph without any loops or multiple edges. It's a clean, simple representation of relationships.

The points in a graph that represent individual entities or elements.

A connected graph is Eulerian if and only if every vertex of the graph has an even degree, meaning each vertex has an even number of edges connected to it.

A graph that contains an Euler path, meaning a path that passes through each edge exactly once, has exactly two vertices with an odd degree, while all other vertices have even degree.

A journey or path that begins and ends at the same point and traverses each edge (road or track) exactly once. An Eulerian graph.

If a graph has an Euler circuit, the graph is considered Eulerian.

A connected graph contains an Euler path if and only if it has exactly two odd vertices.

A graph where every possible edge is drawn between the vertices. This means any two vertices are directly connected.

A graph where vertices can be divided into two groups such that every edge connects vertices from different groups.

A graph where edges have a direction, indicating a flow from one vertex to another.

A graph where every pair of vertices is connected by exactly one path. It contains no loops or cycles.

Graphs that have the same connections between vertices, even if their visual representations differ.

A closed path that starts and ends at the same vertex, traversing each edge once.

A Hamiltonian circuit is a cycle in a graph where every vertex is visited exactly once, and the path starts and ends at the same vertex.

A Hamiltonian path is a path in a graph where every vertex is visited exactly once, but the path starts and ends at different vertices.

A graph is Hamiltonian if it contains a Hamiltonian circuit. It must be possible to travel around the graph visiting every node exactly once and ending where you started.

Weighted graphs have values (weights) assigned to each edge. These weights can represent cost, distance, time, or other relevant factors.

These algorithms (like the shortest path algorithm) aim to find the most efficient Hamiltonian circuit in a weighted graph, considering the weights assigned to each edge.

An algorithm that aims to find the cheapest solution possible at each step, considering only local information.

A method to find a Hamiltonian circuit in a weighted graph, where the algorithm chooses the least costly edge at each step.

An algorithm that helps in finding the cheapest (or most efficient) Hamiltonian circuit in a complete graph.

A method for finding a Hamiltonian circuit in a complete graph that sequentially selects the edges with the smallest weights, prioritizing edges that connect vertices with the least number of connected edges already.

The total weight of all edges in a Hamiltonian circuit.

Creating a Hamiltonian circuit where the selected edges have the smallest weight, potentially resulting in a suboptimal solution.

Creating a Hamiltonian circuit by systematically connecting vertices with the least connected edges, potentially leading to a less efficient solution.

The edge-picking algorithm might provide a more efficient route compared to the greedy algorithm because it considers the overall connectivity of the graph.

The choice of algorithm for finding a Hamiltonian circuit depends on the specific application and the desired outcome.

A graph that can be drawn on a plane without any edges crossing, except at vertices.

A drawing of a planar graph where no edges intersect except at vertices.

The smallest number of colors needed to color the vertices of a graph so that no two adjacent vertices have the same color.

A graph where every vertex can be colored using a specific number of colors such that no two adjacent vertices have the same color.

The process of assigning colors to vertices in a graph such that no two adjacent vertices have the same color.

In graph theory, it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints.

A graph where all edges are connected to neighboring countries, representing a map, is always planar because it can be drawn on a plane without any edges crossing.

The Four-Color Theorem states that every planar graph (a graph that can be drawn on a plane without any edges crossing) can be colored with a maximum of four colors so that no two adjacent vertices have the same color.

A graph that requires more than four colors to color its vertices without adjacent (connected) vertices having the same color is a non-planar graph. This means it cannot be drawn on a plane without edges crossing.

The chromatic number of a graph is the minimum number of colors required to color the vertices so that no two adjacent vertices have the same color. In general, finding this number is computationally difficult.

It is a way to represent a map as a graph, where vertices represent countries, edges represent shared boundaries, and countries touching only at a corner are not considered neighbors. This graph accurately reflects the geographical relationships for coloring purposes.

Determining if a graph can be colored with just two colors (red and blue) without having adjacent vertices share the same color. This helps understand if simple patterns can be applied for coloring.

A system of arithmetic where numbers are represented as remainders after division by a fixed number called the modulus.

An operation in modular arithmetic that represents adding numbers on a 12-hour clock, where 12:00 is equivalent to 0.

An operation in modular arithmetic that represents subtracting numbers on a 12-hour clock, where 12:00 is equivalent to 0.

A statement indicating that two integers have the same remainder when divided by a specific natural number (modulus).

The natural number by which we are dividing in modular arithmetic. It determines the size of the 'cycle' in the system.

The process of adding or subtracting numbers within a specific modular system, taking into account the modulus to determine the remainder.

Using modular arithmetic to calculate the day of the week for a specific date, considering the number of days in a week (modulus 7).

Applying the concept of modular arithmetic to real-world scenarios like calculating time, days of the week, and repeating cycles.

A mathematical operation where you are only concerned with the remainder after division.

When you add two numbers modulo 'n,' you add them normally and then divide by 'n,' keeping only the remainder as the result.

When you subtract two numbers modulo 'n,' you subtract them normally and then divide by 'n,' keeping only the remainder as the result.

When you multiply two numbers modulo 'n,' you multiply them normally and then divide by 'n,' keeping only the remainder as the result.

When you divide two numbers modulo 'n,' you divide them normally and then multiply the result by 'n,' keeping only the remainder as the result.

A mathematical operation that finds the remainder after division by a specific number (modulus). In our example, it helps us calculate the day of the week after a certain number of days.

A method used to track the days of the week. Each day is assigned a numeric value, starting with Monday as 1 and Sunday as 7.

The number of days in a year, typically 365, with an extra day added (366) in a leap year.

A year that occurs every four years, with 366 days. This extra day is added to February, making it 29 days long.

The day of the week that occurs after a specific number of days from a given starting day. You can use day-of-the-week arithmetic and the modulo operation to calculate it.

When calculating the day of the week after a certain period, consider leap years, which add one extra day, affecting the result.

The concept that the days of the week repeat in a cycle of seven days.

The final day of the week after a specific number of days from a given starting day.

A 13-digit number assigned to books, used for identification and error checking. The first 12 digits represent information about the book, while the last digit is the check digit.

The last digit of an ISBN, calculated using a specific formula to ensure the accuracy of the ISBN. It acts as a check to detect errors.

A 12-digit code used for identifying products, primarily used in grocery stores.

A year that does not have an extra day, with 365 days.

A formula used to calculate the ISBN check digit, based on modular arithmetic. It involves multiplying each digit of the ISBN by a specific weight and then finding the modulus of the sum.

A formula used to calculate the check digit for a UPC, ensuring its accuracy. It involves adding and subtracting digits, then taking the remainder after dividing by 10.

A series of steps used to check the validity of a credit card number. It involves doubling every other digit, adding the resulting numbers, and checking if the sum is divisible by 10.

The final digit in a UPC or credit card number that acts as a verification tool. It's calculated using a specific formula or algorithm.

A technique used to determine the day of the week for a specific date. It involves using modular arithmetic with a modulus of 7, representing the days of the week.

The study of making and breaking secret codes.

A message before it is encoded or encrypted.

A message after it has been encoded or encrypted.

The process of changing plaintext into ciphertext using a secret code.

A code where each letter of the alphabet is shifted the same number of positions to the right.

The process of changing ciphertext into plaintext.

A numerical equivalent for each letter in the English alphabet. A=1, B=2, C=3, ... Z=26

The process of substituting each letter in plaintext with the letter that is a specific number of letters after it in the alphabet.