1. **Determine which graphs of these families are the same:** - $N_1 = K_1$ because $N_1$ is a null graph with 1 vertex and no edges, which is the same as the complete graph $K_1$. - $K_2 = P_2$ because $K_2$ is a complete graph with 2 vertices connected by an edge, and $P_2$ is a path graph with 2 vertices connected by an edge. 2. **Find the order and size of each graph family in general:** - $N_n$ (Null graph): order = $n$, size = $0$ (no edges). - $K_n$ (Complete graph): order = $n$, size = $\frac{n(n-1)}{2}$ (all possible edges). - $P_n$ (Path graph): order = $n$, size = $n-1$ (edges form a single path). - $C_n$ (Cycle graph): order = $n$, size = $n$ (edges form a cycle). - $W_n$ (Wheel graph): order = $n$, size = $2(n-1)$ (a cycle of $n-1$ vertices plus a center vertex connected to all others). 3. **The Petersen graph and each wheel are how much partite graphs?** - Petersen graph is $3$-partite (it is a well-known $3$-partite graph). - Wheel graph $W_n$ is $3$-partite if $n$ is odd, and $2$-partite if $n$ is even. 4. **Find the diameter of each of these graphs in general:** - $N_n$: diameter is undefined or $\infty$ since no edges. - $K_n$: diameter = $1$ (all vertices connected). - $P_n$: diameter = $n-1$ (longest path). - $C_n$: diameter = $\lfloor \frac{n}{2} \rfloor$ (half the cycle length). - $W_n$: diameter = $2$ (center vertex connects all others). - Petersen graph: diameter = $2$. 5. **Vertices of graph $G$ are integers, two vertices $a,b$ adjacent if $a+b$ is odd. To which well-known class does $G$ belong?** - This graph is a bipartite graph because adjacency depends on parity: vertices can be split into even and odd integers, and edges only connect vertices from different parity sets. **Final answers:** - $N_1 = K_1$, $K_2 = P_2$. - Orders and sizes as above. - Petersen graph is $3$-partite; wheel graphs are $2$- or $3$-partite depending on $n$. - Diameters as above. - $G$ is a bipartite graph.