Respond to the classmate who published their initial post right above yours. If you posted first, then respond to the last classmate's post on the discussion board.$$

In your response, answer the following questions:

Did your classmate answer all questions correctly? If not, identify and correct the mistakes.Use your classmate$$s first tree in Part $1$ and list the vertices using in$-$order traversal.Use your classmate$$s second tree in Part $1$ and list the vertices using post$-$order traversal.Use your classmate$$s first tree in Part $1$ and list the vertices using pre$-$order traversal.Find the degree of each vertex of your classmate$$s graph Km$,$n$.$

Classmate Response:

Hello class,$$

Using my name, Larissa Allen, the longer name is Larissa $(7$ characters$).$

Pre$-$order Traversal$$

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In$-$order Traversal$$

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Post$-$order Traversal$$

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Part $2$: Graph Questions$$

Edges in Km$,$nK$\_\{$m$,$n$\}$Km$,$n

n$=7$n $=7$n$=7($first name length$),$ m$=5$m $=5$m$=5($last name length$).$

The number of edges in Km$,$nK$\_\{$m$,$n$\}$Km$,$n$$ is m$$n$=57=35$m $\backslash$times n $=5\backslash$times $7=35$m$$n$=57=35.$

To create Km$,$nK$\_\{$m$,$n$\}$Km$,$n$,$ use vertices numbered $111$ to $121212($total m$+$n$=12$m $+$ n $=12$m$+$n$=12).$ Assign $111$ to $555$ to one set and $666$ to $121212$ to the other set.$$

Edges in Kn$+$mK$\_\{$n$+$m$\}$Kn$+$m

n$+$m$=12$n $+$ m $=12$n$+$m$=12.$

Kn$+$mK$\_\{$n$+$m$\}$Kn$+$m$$ has n$+$m$2($n$+$m$1)=12112=66\backslash$frac$\{$n$+$m$\}\{2\}\backslash$times $($n$+$m$-1)=\backslash$frac$\{12\backslash$times $11\}\{2\}=662$n$+$m$($n$+$m$1)=21211=66$ edges.$$

Edges in Wm$+$nW$\_\{$m$+$n$\}$Wm$+$n

Wm$+$nW$\_\{$m$+$n$\}$Wm$+$n$$ has $($m$+$n$)+($m$+$n$1)($m$+$n$2)2($m$+$n$)+\backslash$frac$\{($m$+$n$-1)($m$+$n$-2)\}\{2\}($m$+$n$)+2($m$+$n$1)($m$+$n$2)$ edges.$$

Plugging in m$+$n$=12$m$+$n $=12$m$+$n$=12,$ we get $12+11102=12+55=6712+\backslash$frac$\{11\backslash$times $10\}\{2\}=12+55=6712+21110=12+55=67$ edges.