1. **Given the graph $G$ with vertices $v_1$ to $v_8$ and specified edges:** - **Vertex set $V$:** The set of all vertices is $$V = \{v_1, v_2, v_3, v_4, v_5, v_6, v_7, v_8\}.$$ - **Edge set $E$:** Based on the connections described, each edge is undirected and listed once: $$E = \{(v_1,v_2), (v_1,v_3), (v_1,v_4), (v_2,v_5), (v_3,v_5), (v_4,v_5), (v_5,v_6), (v_5,v_7), (v_5,v_8)\}.$$ 2. **Degrees of each vertex:** - Degree of $v_1$: connected to $v_2,v_3,v_4$ is $3$ - Degree of $v_2$: connected to $v_1,v_5$ is $2$ - Degree of $v_3$: connected to $v_1,v_5$ is $2$ - Degree of $v_4$: connected to $v_1,v_5$ is $2$ - Degree of $v_5$: connected to $v_2,v_3,v_4,v_6,v_7,v_8$ is $6$ - Degree of $v_6$: connected to $v_5$ is $1$ - Degree of $v_7$: connected to $v_5$ is $1$ - Degree of $v_8$: connected to $v_5$ is $1$ 3. **Order and size of the graph:** - Order = number of vertices = $|V| = 8$ - Size = number of edges = $|E| = 9$ --- 4. **Party graph problem:** - There are 10 people numbered 1 to 10. - Given number known by persons 2 through 10: 2 → 7, 3 → 3, 4 → 4, 5 → 5, 6 → 5, 7 → 4, 8 → 3, 9 → 2, 10 → 1 - The graph is undirected, so sum of degrees must be even. - Sum known excluding person 1: $$7 + 3 + 4 + 5 + 5 + 4 + 3 + 2 + 1 = 34.$$ - Let $x$ = number of people person 1 knows. - Total sum degrees = $x + 34$ must be even. - $34$ is even, so $x$ must be even. - Possible $x$ values range from $0$ to $9$. - Hence, person 1 knows $x$ people with $x \\in \{0, 2, 4, 6, 8\}$ (even numbers only). - Without more constraints, the exact number cannot be uniquely determined. --- 5. **Flow chart program sequencing:** - Vertices: $A,B,C,D,E,F,G,H$ - Edges: $A \to B$, $B \to C,D,E$, $C \to E$, $D \to F$, $E \to F$, $F \to G$, $G \to B,H$ 6. **Shortest execution sequence from $A$ to $H$:** - Key path to reach $H$ minimizing steps. - One shortest path: $$A \to B \to E \to F \to G \to H.$$ - Length: $6$ vertices. 7. **Possible execution sequence with 15 vertices (steps):** - The cycle $B \to C,D,E,F,G \to B$ can be repeated. - Example sequence: $$A, B, C, E, F, G, B, D, F, G, B, C, E, F, G, H$$ (16 vertices, remove one step to get 15) - A 15-step valid sequence could be: $$A, B, C, E, F, G, B, D, F, G, B, C, E, F, G$$ then from $G$ to $H$ final step adding 16th vertex, so remove one repetition: $$A, B, C, E, F, G, B, D, F, G, B, C, E, F, G, H$$ is 16 steps, thus without last $G$, 15 steps do not end at $H$. - Instead, a 15-step path ending at $H$: $$A, B, E, F, G, B, D, F, G, B, C, E, F, G, H.$$ --- **Final answers:** - Vertex set $V = \{v_1,v_2,v_3,v_4,v_5,v_6,v_7,v_8\}$. - Edge set $E = \{(v_1,v_2),(v_1,v_3),(v_1,v_4),(v_2,v_5),(v_3,v_5),(v_4,v_5),(v_5,v_6),(v_5,v_7),(v_5,v_8)\}$. - Degrees: $deg(v_1)=3$, $deg(v_2)=2$, $deg(v_3)=2$, $deg(v_4)=2$, $deg(v_5)=6$, $deg(v_6)=1$, $deg(v_7)=1$, $deg(v_8)=1$. - Order $=8$, Size $=9$. - Person 1 knows an even number of people $x \in \{0, 2, 4, 6, 8\}$. - Shortest flow execution length $=6$. - Example flow execution sequence of length 15: $$A, B, E, F, G, B, D, F, G, B, C, E, F, G, H.$$