Graphs

9.1 Find all graphs with 2, 3 and 4 nodes.

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9.2 (a) Is there a graph on 6 nodes with degrees 2, 3, 3, 3, 3, 3?

(b) Is there a graph on 6 nodes with degrees 0, 1, 2, 3, 4, 5?

(c) How many graphs are there on 4 nodes with degrees 1, 1, 2, 2?

(d) How many graphs are there on 10 nodes with degrees 1, 1, 1, 1, 1, 1, 1, 1, 1, 1?

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9.3 At the end of the party, everybody knows everybody else. Draw the graph representing this situation. How many edges does it have?

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9.4 Draw the graphs representing the bonds between atoms in

(a) a water molecule;

(b) a methane molecule;

(c) two water molecules.

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9.5 (a) Draw a graph with nodes representing the numbers $1,2,\dots ,10$, in which two nodes are connected by an edge if and only if one is a divisor of the other.

(b) Draw a graph with nodes representing the numbers $1,2,\dots ,10$, in which two nodes are connected by an edge if and only if they have no common divisor larger than 1.

(c) Find the number of edges and the degrees in these graphs, and check that theorem 9.1 holds.

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9.6 What is the largest number of edges a graph with 10 nodes can have?

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9.7 How many graphs are there on 20 nodes? (To make this question precise, we have to make sure we know what it means that two graphs are the same. For the purposes of this exercise, we consider the nodes given, and labeled say, as Alice, Bob, … The graph consisting of a single edge connecting Alice and Bob is different from the graph consisting of a single edge connecting Eve and Frank.)

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9.8 Formulate the following assertion as a theorem about graphs, and prove it: At every party one can find two people who know the same number of other people (like Bob and Eve in our first example).

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9.9 Find all complete graphs, paths and cycles among the graphs in the previous figures.

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9.10 How many subgraphs does an empty graph on $n$ nodes have? How many subgraphs does a triangle have?

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9.11 Is the proof as given above valid if

(a) the node $a$ lies on the path $Q$;

(b) the paths $P$ and $Q$ have no node in common except $b$.

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9.12 (a) We delete an edge $e$ from a connected graph $G$. Show by an example that the remaining graph may not be connected.

(b) Prove that if we assume that the deleted edge $e$ belongs to a cycle that is a subgraph of $G$, then the remaining graph is connected.

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9.13 Let $G$ be a graph and let $u$ and $v$ be two nodes of $G$. A walk in $G$ from $u$ to $v$ is a sequence of nodes $(w_0,w_1,\dots ,w_k)$ such that $w_0=u$, $w_k=v$, and consecutive nodes are connected by an edge, i.e., $w_i{w}_{i+1}\in E$ for $i=0,1,\dots ,k-1$.

(a) Prove that if there is a walk in $G$ from $u$ to $v$, then $G$ contains a path connecting $u$ to $v$.

(b) Use part (a) to give another proof of the fact that if $G$ contains a path connecting $a$ and $b$, and also a path connecting $b$ to $c$, then it contains a path connecting $a$ to $c$.

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9.14 Let $G$ be a graph, and let $H_1=(V_1,E_1)$ and $H_2=(V_2,E_2)$ be two subgraphs of $G$ that are connected. Assume that $H_1$ and $H_2$ have at least one node in common. Form their union, i.e., the subgraph $H=(V^{\prime },E^{\prime })$, where $V^{\prime }=V_1\cup V_2$ and $E^{\prime }=E_1\cup E_2$. Prove that $H$ is connected.

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9.15 Determine the connected components of the graphs constructed in exercise 9.1.

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9.16 Prove that no edge of $G$ can connect nodes in different connected components.

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9.17 Prove that a node $v$ is a node of the connected component of $G$ containing node $u$ if and only if $G$ contains a path connecting $u$ to $v$.

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9.18 Prove that a graph with $n$ nodes and more than $\left(\genfrac{}{}{0ex}{}{n-1}{2}\right)$ edges is always connected.

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