Find $($draw$)$ all non$-$isomorphic graphs $($i$.$e$.$ unlabelled graphs$)$ with $6$ vertices and $6$ edges for which the

maximum degree of a vertex is equal to $4.$

$2.$ A$.$ How many subgraphs of Kn are isomorphic to K$3,5?$

B$.$ How many subgraphs of Kn$,$n are isomorphic to K$3,5?$

$3.$ Let n $>=$ t and let K$$

t be a graph obtained from Kt by deleting one edge. How many subgraphs of Kn are

isomorphic to K$$

t $?$

$4.$ Let H be a graph obtained from K$6$ by deleting the three edges of a cycle of length $3.$ How many subgraphs

of Kn are isomorphic to H$?$

$5.$ How many cycles on k vertices are there in the complete graph Kn$?$

Hint: Consider choosing a sequence of k distinct vertices, say v$1,$ v$2,...,$ vk from the vertices of Kn$,$ say

v$1,$ v$2,...,$ vk and determine the corresponding cycle with edges $\{$vi$,$ vi$+1\}$ for $1<=$ i $<=$ n $1$ and $\{$vn$,$ v$1\}.$

This question amounts to find the number of ways you can produce the same cycle.

$6.$ Draw all subgraphs of K$3.$

$7.$ Let n $>=$ a$,$ b $>=1.$

A$.$ How many subgraphs of Kn are isomorphic to Ka$,$b if a $6=$ b$.$

B$.$ How many subgraphs of Kn are isomorphic to Ka$,$b if a $=$ b$.$

$8.$ Draw two non$-$isomorphic graphs with $6$ vertices where each vertex has degree $2.$

$9.$ Up to isomorphism, find $($draw$)$ all graphs with $5$ vertices and $3$ edges.

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