### Analyzing Euler Circuits and Trails in Graph TheoryIn this section, we will explore the criteria for Euler circuits and Euler trails within two common types of graphs: cycle graphs (Cₙ) and complete graphs (Kₙ).#### Questions to Consider:**(a)** For which values of \( n \) does \( C_n \) have an Euler circuit?- **Explanation**: An Euler circuit exists in a graph if it is connected and every vertex has an even degree. For cycle graphs \( C_n \), every vertex in the cycle has a degree of 2. Therefore, \( C_n \) will have an Euler circuit for all values of \( n \geq 3 \).**(b)** For which values of \( n \) does \( C_n \) have an Euler trail?- **Explanation**: An Euler trail exists if a graph is connected and has exactly two vertices of odd degree. In \( C_n \), since every vertex has an even degree, the graph will have an Euler trail only when the conditions for an Euler circuit are satisfied. Thus, \( C_n \) will have an Euler trail for all values of \( n \geq 3 \); however, this is the same condition as having an Euler circuit.**(c)** For which values of \( n \) does \( K_n \) have an Euler circuit?- **Explanation**: The complete graph \( K_n \) has an Euler circuit if it is connected and all vertices have even degree. In \( K_n \), each vertex has degree \( n-1 \). Thus, \( K_n \) will have an Euler circuit when \( n-1 \) is even, which occurs when \( n \) is odd (\( n = 2k + 1 \), where \( k \) is an integer).**(d)** For which values of \( n \) does \( K_n \) have an Euler trail?- **Explanation**: For \( K_n \) to have an Euler trail, precisely two vertices must have an odd degree, and the rest must have even degrees. Since each vertex in \( K_n \) has degree \( n-1 \), which is odd for even \( n \), all vertices have odd degrees. Hence, no Euler trail exists for any \( n \).As you explore these concepts, consider how the degree of vertices plays a crucial role in

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Answer each question below and justify your answer. (a) For which values of n does Cn have an Euler circuit? (b) For which values of n does Cn have an Euler trail? (c) For which values of n does Kn have an Euler circuit? (d) For which values of n does K, have an Euler trail?

Answer each question below and justify your answer. (a) For which values of n does Cn have an Euler circuit? (b) For which values of n does Cn have an Euler trail? (c) For which values of n does Kn have an Euler circuit? (d) For which values of n does K, have an Euler trail?

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Fast Fourier Transform Concepts

The algorithm used to convert a sequence to its Discrete Fourier transform (DFT) is known as the Fast Fourier transform. It can be used for both DFT and IDFT (Inverse DFT) transformation.

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### Analyzing Euler Circuits and Trails in Graph Theory

In this section, we will explore the criteria for Euler circuits and Euler trails within two common types of graphs: cycle graphs (Cₙ) and complete graphs (Kₙ).

#### Questions to Consider:

**(a)** For which values of \( n \) does \( C_n \) have an Euler circuit?

- **Explanation**: An Euler circuit exists in a graph if it is connected and every vertex has an even degree. For cycle graphs \( C_n \), every vertex in the cycle has a degree of 2. Therefore, \( C_n \) will have an Euler circuit for all values of \( n \geq 3 \).

**(b)** For which values of \( n \) does \( C_n \) have an Euler trail?

- **Explanation**: An Euler trail exists if a graph is connected and has exactly two vertices of odd degree. In \( C_n \), since every vertex has an even degree, the graph will have an Euler trail only when the conditions for an Euler circuit are satisfied. Thus, \( C_n \) will have an Euler trail for all values of \( n \geq 3 \); however, this is the same condition as having an Euler circuit.

**(c)** For which values of \( n \) does \( K_n \) have an Euler circuit?

- **Explanation**: The complete graph \( K_n \) has an Euler circuit if it is connected and all vertices have even degree. In \( K_n \), each vertex has degree \( n-1 \). Thus, \( K_n \) will have an Euler circuit when \( n-1 \) is even, which occurs when \( n \) is odd (\( n = 2k + 1 \), where \( k \) is an integer).

**(d)** For which values of \( n \) does \( K_n \) have an Euler trail?

- **Explanation**: For \( K_n \) to have an Euler trail, precisely two vertices must have an odd degree, and the rest must have even degrees. Since each vertex in \( K_n \) has degree \( n-1 \), which is odd for even \( n \), all vertices have odd degrees. Hence, no Euler trail exists for any \( n \).

As you explore these concepts, consider how the degree of vertices plays a crucial role in

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