Introductory Combinatorics

Richard A. Brualdi

Chapter 11

Introduction to Graph Theory - all with Video Answers

Educators

Chapter Questions

Problem 1

How many nonisomorphic graphs of order 1 are there? of order $2 ?$ of order 3? Explain why the answer to each of the preceding questions is $\infty$ for general graphs.

Chris Trentman

Chris Trentman

Numerade Educator

Problem 2

Determine each of the 11 nonisomorphic graphs of order 4, and give a planar representation of each.

Melissa Salvador

Melissa Salvador

Numerade Educator

Problem 3

Does there exist' a graph of order 5 whose degree sequence equals $(4,4,3,2,2)$ ?

Sheryl Ezze

Numerade Educator

Problem 4

Does there exist a graph of order 5 whose degree sequence equals $(4,4,4,2,2) ?$ a multigraph?

Chris Trentman

Numerade Educator

Problem 5

Use the pigeonhole principle to prove that a graph of order $n \geq 2$ always has two vertices of the same degree. Does the same conclusion hold for multigraphs?

Prathan Jarupoonphol

Prathan Jarupoonphol

Numerade Educator

Problem 6

Let $\left(d_{1}, d_{2}, \ldots, d_{n}\right)$ be a sequence of $n$ nonnegative even integers. Prove that there exists a general graph with this sequence as its degree sequence.

Julian Wong

Numerade Educator

Problem 7

Let $\left(d_{1}, d_{2}, \ldots, d_{n}\right)$ be a sequence of $n$ nonnegative integers whose sum $d_{1}+d_{2}+$ $\cdots+d_{n}$ is even. Prove that there exists a general graph with this sequence as its degree sequence. Devise an algorithm to construct such a general graph.

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 8

Let $G$ be a graph with degree sequence $\left(d_{1}, d_{2}, \ldots, d_{n}\right) .$ Prove that, for each $k$ with $0<k<n$,
$$
\sum_{i=1}^{k} d_{i} \leq k(k-1)+\sum_{i=k+1}^{n} \min \left\{k, d_{i}\right\} .
$$

Brian Lin

Numerade Educator

Problem 9

Draw a connected graph whose degree sequence equals
$$
(5,4,3,3,3,3,3,2,2)
$$

Wendi Zhao

Numerade Educator

Problem 10

Prove that any two connected graphs of order $n$ with degree sequence $(2,2, \ldots, 2)$ are isomorphic.

Chris Trentman

Numerade Educator

Problem 11

Determine which pairs of the general graphs in Figure $11.39$ are isomorphic and, if isomorphic, find an isomorphism.

Chris Trentman

Numerade Educator

Problem 12

Determine which pairs of the graphs in Figure $11.40$ are isomorphic, and for those that are isomorphic, find an isomorphism.

Chris Trentman

Numerade Educator

Problem 13

Prove that, if two vertices of a general graph are joined by a walk, then they are joined by a path.

Prathan Jarupoonphol

Prathan Jarupoonphol

Numerade Educator

Problem 14

Let $x$ and $y$ be vertices of a general graph, and suppose that there is a closed walk containing both $x$ and $y$. Must there be a closed trail containing both $x$ and $y$ ?

Brian Lin

Numerade Educator

Problem 15

Let $x$ and $y$ be vertices of a general graph, and suppose that there is a closed trail containing both $x$ and $y$. Must there be a cycle containing both $x$ and $y$ ?

Brian Lin

Numerade Educator

Problem 16

Let $G$ be a connected graph of order 6 with degree sequence $(2,2,2,2,2,2)$.
(a) Determine all the nonisomorphic induced subgraphs of $G$.
(b) Determine all the nonisomorphic spanning subgraphs of $G$.
(b) Determine all the nonisomorphic subgraphs of order 6 of $G$.

Chris Trentman

Numerade Educator

Problem 17

First, prove that any two multigraphs $G$ of order 3 with degree sequence $(4,4,4)$ are isomorphic. Then
(a) Determine all the nonisomorphic induced subgraphs of $G$.
(b) Determine all the nonisomorphic spanning subgraphs of $G$.
(b) Determine all the nonisomorphic subgraphs of order 3 of $G$.

Anthony Ramos

Numerade Educator

Problem 18

Let $\gamma$ be a trail joining vertices $x$ and $y$ in a general graph. Prove that the edges of $\gamma$ can be partitioned so that one part of the partition determines a path joining $x$ and $y$ and the other parts determine cycles.

Nick Johnson

Numerade Educator

Problem 19

Let $G$ be a general graph and let $G^{\prime}$ be the graph obtained from $G$ by deleting all loops and all but one copy of each edge with multiplicity greater than 1. Prove that $G$ is connected if and only if $G^{\prime}$ is connected. Also prove that $G$ is planar if and only if $G^{\prime}$ is planar.

Victoria Dollar

Victoria Dollar

Numerade Educator

Problem 20

Prove that a graph of order $n$ with at least
$$
\frac{(n-1)(n-2)}{2}+1
$$
edges must be connected. Give an example of a disconnected graph of order $n$ with one fewer edge.

Brian Lin

Numerade Educator

Problem 21

Let $G$ be a general graph with exactly two vertices $x$ and $y$ of odd degree. Let $G^{*}$ be the general graph obtained by putting a new edge $\{x, y\}$ joining $x$ and $y .$ Prove that $G$ is connected if and only if $G^{*}$ is connected.

Brian Lin

Numerade Educator

Problem 22

(This and the following two exercises prove Theorem 11.1.3.) Let $G=(V, E)$ be a general graph. If $x$ and $y$ are in $V$, define $x \sim y$ to mean that either $x=y$ or there is a walk joining $x$ and $y$. Prove that, for all vertices $x, y$, and $z$, we have
(a) $x \sim x$.
(b) $x \sim y$ if and only if $y \sim x$.
(c) if $x \sim y$ and $y \sim z$, then $x \sim z$.

Chris Trentman

Numerade Educator

Problem 23

(Continuation of Exercise 22.) For each vertex $x$, let
$$
C(x)=\{z: x \sim z\} .
$$
Prove the following:
(i) For all vertices $x$ and $y$, either $C(x)=C(y)$ or else $C(x) \cap C(y)=0 .$ In other words two of the sets $C(x)$ and $C(y)$ cannot intersect unless they are equal.
(ii) If $C(x) \cap C(y)=\emptyset$, then there does not exist an edge joining a vertex in $C(x)$ to a vertex in $C(y)$.

WZ

Wen Zheng

Numerade Educator

Problem 24

(Continuation of Exercise 23.) Let $V_{1}, V_{2}, \ldots, V_{k}$ be the different sets that occur among the $C(x)^{\prime}$ 's. Prove the following:
(i) $V_{1}, V_{2}, \ldots, V_{k}$ form a partition of the vertex set $V$ of $G$.
(ii) The general subgraphs $G_{1}=\left(V_{1}, E_{1}\right), G_{2}=\left(V_{2}, E_{2}\right), \ldots, G_{k}=\left(V_{k}, E_{k}\right)$ of
$G$ induced by $V_{1}, V_{2}, \ldots, V_{k}$, respectively, are connected.
The induced subgraphs $G_{1}, G_{2}, \ldots, G_{k}$ are the connected components of $G$.

Prathan Jarupoonphol

Prathan Jarupoonphol

Numerade Educator

Problem 25

Prove Theorem $11.1 .4$.

Nick Johnson

Numerade Educator

Problem 26

Determine the adjacency matrices of the first and second general graphs in Figure $11.39 .$

Chris Trentman

Numerade Educator

Problem 27

Determine the adjacency matrices of the first and second graphs in Figure $11.40 .$

Shubh Ashish

Numerade Educator

Problem 28

Let $A$ and $B$ be two $n$ -by $-n$ matrices of numbers whose entries are denoted by $a_{i j}$ and $b_{i j},(1 \leq i, j \leq n)$, respectively. Define the product $A \times B$ to be the $n$ -by-n matrix $C$ whose entry $c_{i j}$ in row $i$ and column $j$ is given by
$$
c_{i j}=\sum_{p=1}^{n} a_{i p} b_{p j}, \quad(1 \leq i, j \leq n)
$$
If $k$ is a positive integer, define
$$
A^{k}=A \times A \times \cdots \times A \quad\left(k A^{\prime} s\right)
$$
Now let $A$ denote the adjacency matrix of a general graph of order $n$ with vertices $a_{1}, a_{2}, \ldots, a_{n} .$ Prove that the entry in row $i$, column $j$ of $A^{k}$ equals the number of walks of length $k$ in $G$ joining vertices $a_{i}$ and $a_{j}$.

WZ

Wen Zheng

Numerade Educator

Problem 29

Determine if the multigraphs in Figure $11.41$ have Eulerian trails (closed or open). In case there is an Eulerian trail, use the algorithms prsented in this chapter to construct one.

Clarissa Noh

Numerade Educator

Problem 30

Which complete graphs $K_{n}$ have closed Eulerian trails? open Eulerian trails?

Victoria Dollar

Victoria Dollar

Numerade Educator

Problem 31

Prove Theorem $11.2 .4$.

Carson Merrill

Numerade Educator

Problem 32

What is the fewest number of open trails into which the edges of GraphBuster can be partitioned?

Chris Trentman

Numerade Educator

Problem 33

Show how, removing pencil from paper the fewest number of times, to trace the plane graphs in Figures $11.15,11.16$, and $11.17 .$

Charlotte Ihme

Numerade Educator

Problem 34

Determine all nonisomorphic graphs of order at most 6 that have a closed Eulerian trail.

Chris Trentman

Numerade Educator

Problem 35

Show how, removing pencil from paper the fewest number of times, to trace out the graph of the regular dodecahedron shown in Figure $11.18 .$

Kara Merfeld

Numerade Educator

Problem 36

Let $G$ be a connected graph. Let $\gamma$ be a closed walk that contains each edge of $G$ at least once. Let $G^{*}$ be the multigraph obtained from $G$ by increasing the multiplicity of each edge from 1 to the number of times it occurs in $\gamma$. Prove that $\gamma$ is a closed Eulerian trail in $G^{*}$. Conversely, suppose we increase the multiplicity of some of the edges of $G$ and obtain a multigraph with $m$ edges, each of whose vertices has even degree. Prove that there is a closed walk in $G$ of length $m$ which contains each edge of $G$ at least once. This exercise shows that the Chinese postman problem for $G$ is equivalent to determining the smallest number of copies of the edges of $G$ that need to be inserted so as to obtain a multigraph all of whose vertices have even degree.

Chris Trentman

Numerade Educator

Problem 37

Solve the Chinese postman problem for the complete graph $K_{6}$.

Erika Bustos

Numerade Educator

Problem 38

Solve the Chinese postman problem for the graph obtained from $K_{6}$ by removing any edge.

Erika Bustos

Numerade Educator

Problem 39

Call a graph cubic if each vertex has degree equal to 3 . The complete graph $K_{4}$ is the smallest example of a cubic graph. Find an example of a connected, cubic graph that does not have a Hamilton path.

WZ

Wen Zheng

Numerade Educator

Problem 40

-=Let $G$ be a graph of order $n$ having at least
$$
\frac{(n-1)(n-2)}{2}+2
$$
edges, Prove that $G$ has a Hamilton cycle. Exhibit a graph of order $n$ with one fewer edge that does not have a Hamilton cycle.

Carson Merrill

Numerade Educator

Problem 41

Let $n \geq 3$ be an integer. Let $G_{n}$ be the graph whose vertices are the $n !$ permutations of $\{1,2, \ldots, n\}$, wherein two permutations are joined by an edge if and only if one can be obtained from the other by the interchange of two numbers (an arbitrary transposition). Deduce from the results of Section $4.1$ that $G_{n}$ has a Hamilton cycle.

WZ

Wen Zheng

Numerade Educator

Problem 42

Prove Theorem $11.3 .4$.

Nick Johnson

Numerade Educator

Problem 43

Devise an algorithm analogous to our algorithm for a Hamilton cycle that constructs a Hamilton path in graphs satisfying the condition given in Theorem $11.3 .4$

Carson Merrill

Numerade Educator

Problem 44

Which complete bipartite graphs $K_{m, n}$ have Hamilton cycles? Which have Hamilton paths?

Carson Merrill

Numerade Educator

Problem 45

Prove that a multigraph is bipartite if and only if each of its connected components is bipartite.

Chris Trentman

Numerade Educator

Problem 46

Prove that $K_{m, n}$ is isomorphic to $K_{n, m}$.

Abid Hussain

Numerade Educator

Problem 47

Prove that a bipartite multigraph with an odd number of vertices does not have a Hamilton cycle.

Carson Merrill

Numerade Educator

Problem 48

Is GraphBuster a bipartite graph? If so, find a bipartition of its vertices. What. if we delete the loops?

Carson Merrill

Numerade Educator

Problem 49

Let $V=\{1,2, \ldots, 20\}$ be the set of the first 20 positive integers. Consider the graphs whose vertex set is $V$ and whose edge sets are defined below. For earth graph, investigate whether the graph (i) is connected (if not connected, determin" the connected components), (ii) is bipartite, (iii) has an Eulerian trail, and (iv) has a Hamilton path.
(a) $\{a, b\}$ is an edge if and only if $a+b$ is even.
(b) $\{a, b\}$ is an edge if and only if $a+b$ is odd.
(c) $\{a, b\}$ is an edge if and only if $a \times b$ is even.
(d) $\{a, b\}$ is an edge if and only if $a \times b$ is odd.
(e) $\{a, b\}$ is an edge if and only if $a \times b$ is a perfect square.
(f) $\{a, b\}$ is an edge if and only if $a-b$ is divisible by 3 .

Brian Lin

Numerade Educator

Problem 50

What is the smallest number of edges that can be removed from $K_{5}$ to leave a bipartite graph?

Harry Evans

Numerade Educator

Problem 51

Find a knight's tour on the boards of the following sizes:
(a) $5-\mathrm{by}-5$
(b) 6 -by-6
(c) 7 -by - 7

Carole Wastog

Numerade Educator

Problem 52

$*$ Prove that there does not exist a knight's tour on a 4 -by-4 board.

Carson Merrill

Numerade Educator

Problem 53

Prove that a graph is a tree if and only if it does not contain any cycles, but the insertion of any new edge always creates exactly one cycle.

Victoria Dollar

Victoria Dollar

Numerade Educator

Problem 54

Which trees have an Eulerian path?

Victoria Dollar

Victoria Dollar

Numerade Educator

Problem 55

Which trees have a Hamilton path?

Carson Merrill

Numerade Educator

Problem 56

Grow all the nonisomorphic trees of order $7 .$

Michelle Nguyen

Michelle Nguyen

Numerade Educator

Problem 57

Let $\left(d_{1}, d_{2}, \ldots, d_{n}\right)$ be a sequence of integers.
(a) Prove that there is a tree of order $n$ with this degree sequence if and only if $d_{1}, d_{2}, \ldots, d_{n}$ are positive integers with sum $d_{1}+d_{2}+\cdots+d_{n}=2(n-1)$.
(b) Write an algorithm that, starting with a sequence $\left(d_{1}, d_{2}, \ldots, d_{n}\right)$ of positive integers, either constructs a tree with this degree sequence or concludes that none is possible.

AG

Ankit Gupta

Numerade Educator

Problem 58

A forest is a graph each of whose connected components is a tree. In particular, a tree is a forest. Prove that a graph is a forest if and only if it does not have any cycles.

Victoria Dollar

Victoria Dollar

Numerade Educator

Problem 59

Prove that the removal of an edge from a tree leaves a forest of two trees.

Victoria Dollar

Victoria Dollar

Numerade Educator

Problem 60

Let $G$ be a forest of $k$ trees. What is the fewest number of edges that can be inserted in $G$ in order to obtain a tree?

Victoria Dollar

Victoria Dollar

Numerade Educator

Problem 61

Determine a spanning tree for GraphBuster.

Chris Trentman

Numerade Educator

Problem 62

Prove that, if a tree has a vertex of degree $p$, then it has at least $p$ pendent vertices.

Prashant Bana

Numerade Educator

Problem 63

Determine a spanning tree for each of the graphs in Figures $11.15$ through $11.17 .$

Nick Johnson

Numerade Educator

Problem 64

For each integer $n \geq 3$ and for each integer $k$ with $2 \leq k \leq n-1$, construct a tree of order $n$ with exactly $k$ pendent vertices.

WZ

Wen Zheng

Numerade Educator

Problem 65

Use the algorithm for a spanning tree in Section $11.5$ to construct a spanning tree of the graph of the dodecahedron.

Nick Johnson

Numerade Educator

Problem 66

How many cycles does a connected graph of order $n$ with $n$ edges have?

Prathan Jarupoonphol

Prathan Jarupoonphol

Numerade Educator

Problem 67

Let $G$ be a graph of order $n$ that is not necessarily connected. A forest is defined in Exercise 58. A spanning forest of $G$ is a forest consisting of a spanning tree of each of the connected components of $G$. Modify the algorithm for a spanning tree given in Section $11.5$ so that it constructs a spanning forest of $G$.

Victoria Dollar

Victoria Dollar

Numerade Educator

Problem 68

Determine whether the Shannon switching games played on the graphs in Figure $11.42$ are positive, negative, or neutral games.

Nicholas Bondra

Nicholas Bondra

Numerade Educator

Problem 69

Let $G$ be a connected multigraph. An edge-cut of $G$ is a set $F$ of edges whose removal disconnects $G$. An edge-cut $F$ is minimal, provided that no subset of $F$ other than $F$ itself is an edge-cut. Prove that a bridge is always a minimal edge-cut, and conclude that the only minimal edge-cuts of a tree are the sets consisting of a single edge.

Victoria Dollar

Victoria Dollar

Numerade Educator

Problem 70

Let $G$ be a connected multigraph having a vertex of degree $k$. Prove that $G$ has a minimal edge-cut $F$ with $|F| \leq k$.

WZ

Wen Zheng

Numerade Educator

Problem 71

Let $F$ be a minimal edge-cut of a connected multigraph $G=(V, E)$. Prove that there exists a subset $U$ of $V$ such that $F$ is precisely the set of edges that join $a$ vertex in $U$ to a vertex in the complement $\bar{U}$ of $U$.

WZ

Wen Zheng

Numerade Educator

Problem 72

(Continuation of Exercise 71.) Prove that a spanning tree of a connected multigraph contains at least one edge of every edge-cut.

Chris Trentman

Numerade Educator

Problem 73

Use the algorithm for growing a spanning tree in Section $11.7$ in order to grow a spanning tree of GraphBuster. (Note: GraphBuster is a general graph and has loops and edges of multiplicity greater than 1. The loops can be ignored and only one copy of each edge need be considered.)

Nick Johnson

Numerade Educator

Problem 74

Use the algorithm for growing a spanning tree in order to grow a spanning tree of the graph of the regular dodecahedron.

Chris Trentman

Numerade Educator

Problem 75

Apply the BF-algorithm of Section $11.7$ to determine a BFS-tree for the following:
(a) The graph of the regular dodecahedron (any root)
(b) GraphBuster (any root)
(c) A graph of order $n$ whose edges are arranged in a cycle (any root)
(d) A complete graph $K_{n}$ (any root)
(e) A complete bipartite graph $K_{m, n}$ (a left-vertex root, and a right-vertex root)
In each case, determine the breadth-first numbers and the distance of each vertex from the root chosen.

Nick Johnson

Numerade Educator

Problem 76

Apply the DF-algorithm of Section $11.7$ to determine a DFS-tree for (a), (b), (c),
(d), and (e) as in Exercise 75. In each case, determine the depth-first numbers.

Nick Johnson

Numerade Educator

Problem 77

Let $G$ be a graph that has a Hamilton path which joins two vertices $u$ and
$v$. Is the Hamilton path a DFS-tree rooted at $u$ for $G ?$ Could there be other DFS-trees?

Victoria Dollar

Victoria Dollar

Numerade Educator

Problem 78

(Solution of the Chinese postman problem for trees.) Let $G$ be a tree of order $n$. Prove that the length of a shortest closed walk that includes each edge of $G$ at least once is $2(n-1) .$ Show how the depth-first algorithm finds a walk of length $2(n-1)$ that includes each edge exactly twice.

Chris Trentman

Numerade Educator

Problem 79

Use Dijkstra's algorithm in order to construct a distance tree for $u$ for the weighted graph in Figure $11.43$, with specified vertex $u$ as shown.

WZ

Wen Zheng

Numerade Educator

Problem 80

Consider the complete graph $K_{n}$ with labeled vertices $1,2, \ldots, n$, in which the edge joining vertices $i$ and $j$ is weighted by $c\{i, j\}=i+j$ for all $i \neq j$. Use Dijkstra's algorithm to construct a distance tree rooted at vertex $u=1$ for
(a) $K_{4}$
(b) $K_{6}$
(c) $K_{8}$

Chris Trentman

Numerade Educator

Problem 81

Consider the complete graph $K_{n}$ with labeled vertices $1,2, \ldots, n$, with the weight function $c\{i, j\}=|i-j|$ for all $i \neq j .$ Use Dijkstra's algorithm to construct 8 distance tree rooted at vertex $u=1$ for
(a) $K_{4}$
(b) $K_{6}$
(c) $K_{8}$

Chris Trentman

Numerade Educator

Problem 82

Consider the complete graph $K_{n}$ whose edges are weighted as in Exercise $80 .$ Apply the greedy algorithm to determine a minimum weight spanning tree for
(a) $K_{4}$
(b) $K_{6}$
(c) $K_{8}$

Chris Trentman

Numerade Educator

Problem 83

Consider the complete graph $K_{n}$ whose edges are weighted as in Exercise 81 . Apply the greedy algorithm to determine a minimum weight spanning tree for
(a) $K_{4}$
(b) $K_{6}$
(c) $K_{8}$

Chris Trentman

Numerade Educator

Problem 84

Same as Exercise 82, using Prim's algorithm in place of the greedy algorithm.

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 85

Same as Exercise 83 , using Prim's algorithm in place of the greedy algorithm.

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 86

Let $G$ be a weighted connected graph in which all edge weights are different. Prove that there is exactly one spanning tree of minimum weight.

Chris Trentman

Numerade Educator

Problem 87

Define a caterpillar to be a tree $T$ that has a path $\gamma$ such that every edge of ' $T$ ' is either an edge of $\gamma$ or has one of its vertices on $\gamma$.
(a) Verify that all trees with six or fewer vertices are caterpillars.
(b) Let $T_{7}$ be the tree on seven vertices consisting of three paths of length 2 meeting at a central vertex $c .$ Prove that $T_{7}$ is the only tree on 7 vertices that is not a caterpillar.
(c) Prove that a tree is a caterpillar if and only if it does not contain $T_{7}$ as a spanning subgraph.

Victoria Dollar

Victoria Dollar

Numerade Educator

Problem 88

Let $d_{1}, d_{2}, \ldots, d_{n}$ be positive integers. Prove that there is a caterpillar with degree sequence $\left(d_{1}, d_{2}, \ldots, d_{n}\right)$ if and only if $d_{1}+d_{2}+\cdots+d_{n}=2(n-1)$. Compare with Exercise $57 .$

Julian Wong

Numerade Educator

Problem 89

A graceful labeling of a graph $G$ with vertex set $V$ and with $m$ edges is an injective function $g: V \rightarrow\{0,1,2, \ldots, m\}$ such that the labels $|g(x)-g(y)|$ corresponding to the $m$ edges $\{x, y\}$ of $G$ are $1,2, \ldots, m$ in some order. It has been conjectured by Kotzig and Ringel (1964) that every tree has a graceful labeling. Find a graceful labeling of the tree $T_{7}$ in the previous exercise, any path, and the graph $K_{1, n}$

Chris Trentman

Numerade Educator

Problem 90

Verify that cycles of lengths 5 and 6 cannot be gracefully labeled. Then find graceful labelings of cycles of lengths 7 and 8 .

Amy Jiang

Numerade Educator

Problem 91

Let $G$ be a graph with $n$ vertices $x_{1}, x_{2}, \ldots, x_{n} .$ Let $r_{i}$ be the largest of the distances of $x_{i}$ to the other vertices of $G$. Then
$$
d(G)=\max \left\{r_{1}, r_{2}, \ldots, r_{n}\right\} \text { and } r(G)=\min \left\{r_{1}, r_{2}, \ldots, r_{n}\right\}
$$
are called. respectively, the diameter and radius of $G$. The center of $G$ is the subgraph of $G$ induced by the set of those vertices $x_{i}$ for which $r_{i}=r(G)$, Prove the following assertions:
(a) Determine the radius, diameter, and center of the complete bipartite graph $K_{m, n}$
(b) Determine the radius, diameter, and center of a cycle graph $C_{n} .$
(c) Determine the radius, diameter, and center of a path with $n$ vertices.
(d) Determine the radius, diameter, and center of the graph $Q_{n}$ corresponding to the vertices and edges of an $n$ -dimensional cube.

Brian Lin

Numerade Educator

Problem 92

Prove the following assertions.
(a) The center of a tree $T$ is either a single vertex or two vertices joined by an edge. (Hint: Use induction on the number $n$ of vertices.)
(b) Let $G$ be a graph, and let $\bar{G}$ be the complement graph obtained from $G$ by putting an edge between two vertices of $G$ provided there isn't one in $G$ and removing all edges of $G$. Prove that if $d(G) \geq 3$, then $d(\bar{G}) \leq 3$.

Chris Trentman

Numerade Educator