answer:To satisfy the conditions, we can create a degree sequence as follows: 1. It is graphical: A degree sequence is graphical if it satisfies the Handshaking Lemma (sum of all degrees is even) and the Havel-Hakimi theorem (removing the highest degree vertex and decreasing the degrees of its neighbors results in another graphical sequence). 2. Each vertex has degree between 1 and 4, inclusive. 3. No two vertices have the same degree. Let's create a degree sequence that satisfies these conditions: Degree sequence: 1, 2, 3, 4, 1, 3 This sequence satisfies the Handshaking Lemma since the sum of all degrees is 14, which is even. Now, let's check if it satisfies the Havel-Hakimi theorem: 1. Remove the highest degree vertex (4) and decrease the degrees of its neighbors by 1: New sequence: 0, 1, 2, 0, 2 2. Remove the next highest degree vertex (2) and decrease the degrees of its neighbors by 1: New sequence: 0, 0, 0, 1 3. Remove the next highest degree vertex (1) and decrease the degrees of its neighbors by 1: New sequence: 0, 0, 0 The sequence has been reduced to all zeros, which means it is graphical according to the Havel-Hakimi theorem. Additionally, the degree sequence has each vertex with a degree between 1 and 4, inclusive, and no two vertices have the same degree. Thus, the degree sequence 1, 2, 3, 4, 1, 3 satisfies all the given conditions.

answer:Yes, this degree sequence is possible. We can construct a graph with the given degree sequence as follows: Label the vertices as A, B, C, D, E, F, and G. 1. Connect vertex A (degree 4) to vertices B, C, D, and E. 2. Connect vertex B (degree 2) to vertices A and F. 3. Connect vertex C (degree 2) to vertices A and F. 4. Connect vertex D (degree 3) to vertices A, E, and G. 5. Connect vertex E (degree 3) to vertices A, D, and G. 6. Connect vertex F (degree 2) to vertices B and C. 7. Connect vertex G (degree 4) to vertices D, E, F, and G. The resulting graph has the desired degree sequence {2,2,2,3,3,4,4}: A: connected to B, C, D, E (degree 4) B: connected to A, F (degree 2) C: connected to A, F (degree 2) D: connected to A, E, G (degree 3) E: connected to A, D, G (degree 3) F: connected to B, C, G (degree 3) G: connected to D, E, F (degree 3)

answer:To determine if the degree sequence is graphical or not, we can use the Havel-Hakimi theorem. The Havel-Hakimi theorem states that a degree sequence is graphical if and only if the sequence obtained by deleting the first vertex and subtracting 1 from the next k vertices (where k is the degree of the first vertex) is graphical. Given degree sequence: 3, 3, 3, 2, 2, 1 Sort the sequence in non-increasing order: 3, 3, 3, 2, 2, 1 1. Remove the first vertex (3) and subtract 1 from the next 3 vertices: New sequence: 2, 2, 1, 2, 1 Sort in non-increasing order: 2, 2, 2, 1, 1 2. Remove the first vertex (2) and subtract 1 from the next 2 vertices: New sequence: 1, 1, 1, 1 Sort in non-increasing order: 1, 1, 1, 1 3. Remove the first vertex (1) and subtract 1 from the next vertex: New sequence: 0, 1, 1 Sort in non-increasing order: 1, 1, 0 4. Remove the first vertex (1) and subtract 1 from the next vertex: New sequence: 0, 0 Sort in non-increasing order: 0, 0 The process terminates with a sequence of all zeros, which means the original degree sequence is graphical. Now, let's draw a possible graph with this degree sequence: Label the vertices as A, B, C, D, E, and F with degrees 3, 3, 3, 2, 2, and 1, respectively. 1. Connect vertex A (degree 3) to vertices B, C, and D. 2. Connect vertex B (degree 3) to vertices C and E. 3. Connect vertex C (degree 3) to vertex F. 4. Vertex D (degree 2) is already connected to A and C. 5. Vertex E (degree 2) is already connected to B and C. 6. Vertex F (degree 1) is already connected to C. The resulting graph has the following edges: AB, AC, AD, BC, BE, and CF. This graph has the given degree sequence, so the degree sequence is graphical.

answer:Yes, it is possible to draw such a graph. Here's one possible graph: Label the vertices as A, B, C, D, E, F, G, and H. 1. Connect vertex A to vertices B, C, D, and E (A has a degree of 4). 2. Connect vertex B to vertices C, D, and E (B has a degree of 4). 3. Connect vertex C to vertices F and G (C has a degree of 3). 4. Connect vertex D to vertex H (D has a degree of 2). 5. Connect vertex E to vertex F (E has a degree of 2). 6. Connect vertex F to vertex G (F has a degree of 2). 7. Connect vertex G to vertex H (G has a degree of 1). 8. Vertex H has a degree of 1. The resulting graph has the desired degree sequence {4,4,3,2,2,2,1,1}.