Graph theory

It's just dots and lines

An example

Consider the following graph:

The vertex set is $\{A,B,C,D,E\}$

The edge set is $\{AB, AE, BC, CE, ED\}$

An example

An extremely useful way to represent a graph is called the adjacency matrix. This is the matrix with rows and columns labeled by the vertices (in the same order). The matrix has a $1$ if the row and column indices are adjacent, and zero otherwise.

An example

An example

An example

A Disconnected example

$x_{0} = \begin{bmatrix}1\\0\\0\\0\\0\end{bmatrix}$     or    $x_{0} = \begin{bmatrix}0\\0\\0\\1\\0\end{bmatrix}$

Consider the previous algorithm starting with:

Is it Connected?

The graph has 16 edges, so we could delete each edge, then test again...

Is it Connected?

The Graph Laplacian

For a graph with vertices $v_{1},v_{2},v_{3},\ldots,v_{n}$ the adjacency matrix $A$ is given by

$$A_{i,j} = \begin{cases} 1 & v_i\text{ is adjacent to }v_{j},\\ 0 & \text{otherwise}.\end{cases}$$

The degree of vertex $v_{i}$, denoted $d(v_{i})$ is the number of edges incident with $v_{i}$, that is $$d(v_{j}) = \sum_{i=1}^{n}A_{i,j}.$$

The Laplacian is the $n\times n$ matrix $L$ given by

$$L_{i,j} = \begin{cases} d(v_{i}) & i=j,\\ -A_{i,j} & i\neq j.\end{cases}$$

Consider the previous example:

For any vector $x\in\mathbb{R}^{n}$ we have

$$x^{\top}Lx = [x_{1}\ \ x_{2}\ \ x_{3}\ \ \cdots\ \  x_{n}]\left[\begin{array}{ccccc}L_{1,1} & L_{1,2} & L_{1,3} & \cdots & L_{1,n}\\ L_{2,1} & L_{2,2} & L_{2,3} & \cdots & L_{2,n}\\[-1ex] L_{3,1} & L_{3,2} & L_{3,3} & & \vdots\\ \vdots & \vdots & & \ddots & \vdots\\ L_{n,1} & L_{n,2} & \cdots & \cdots & L_{n,n}\end{array}\right]\left[\begin{array}{c}x_{1}\\ x_{2}\\ x_{3}\\ \vdots \\ x_{n}\end{array}\right]$$

$$= \sum_{i=1}^{n}\sum_{j=1}^{n}x_{i}x_{j}L_{i,j} = \sum_{i=1}^{n}x_{i}^{2}L_{i,i} - \sum_{i=1}^{n}\sum_{j=1}^{n}x_{i}x_{j}A_{i,j}$$

$$\left[\begin{array}{ccccc}d(v_{1}) & -A_{1,2} & -A_{1,3} & \cdots & -A_{1,n}\\ -A_{2,1} & d(v_{2}) & -A_{2,3} & \cdots & -A_{2,n}\\ -A_{3,1} & -A_{3,2} & d(v_{3}) & & \vdots\\ \vdots & \vdots & & \ddots & \vdots\\ -A_{n,1} & -A_{n,2} & \cdots & \cdots & d(v_{n})\end{array}\right]$$

$$= \sum_{i=1}^{n}x_{i}^{2}\sum_{j=1}^{n}A_{i,j} - \sum_{i=1}^{n}\sum_{i=1}^{n}x_{i}x_{j}A_{i,j} = \sum_{i=1}^{n}\sum_{j=1}^{n}(x_{i}^{2}-x_{i}x_{j})A_{i,j}$$

Thus, since $A_{i,j} = A_{j,i}$ we have

$$x^{\top}Lx = \sum_{i=1}^{n}\sum_{j=1}^{n}(x_{i}^{2}-x_{i}x_{j})A_{i,j} = \sum_{i=1}^{n}\sum_{j=1}^{n}(x_{j}^{2}-x_{i}x_{j})A_{i,j}$$

Hence

$$x^{\top}Lx = \frac{1}{2}\sum_{i=1}^{n}\sum_{j=1}^{n}(x_{i}^{2}-2x_{i}x_{j}+x_{j}^{2})A_{i,j} = \frac{1}{2}\sum_{i=1}^{n}\sum_{j=1}^{n}(x_{i}-x_{j})^{2}A_{i,j}.$$

For which vectors $x$ is it the case that

$$x^{\top}Lx = \frac{1}{2}\sum_{i=1}^{n}\sum_{j=1}^{n}(x_{i}-x_{j})^{2}A_{i,j} = 0?$$

Example

For a connected graph:

Instead of $Lx=0$, we look for $x\neq 0$ such that $Lx=\lambda x$ for the smallest $\lambda$.

w/ $\lambda = 0.3033351$

Part 1:

Fundamental Concepts

Graphs

Definition 1.1.2. A graph $G$ is a triple consisting of a vertex set $V(G)$, an edge set $E(G)$, and a relation that associates with each edge two vertices (not necessarily distinct) called its endpoints.

Example. Note that the vertex set is $\{v_{1},v_{2},v_{3},v_{4},v_{5}\}$ and the edge set is $\{e_{1},e_{2},e_{3},e_{4},e_{5}\}$.

The endpoints of $e_{4}$ are $v_{3}$ and $v_{4}$.

Example. The vertex set is $\{v_{1},v_{2},v_{3},v_{4},v_{5}\}$ and the edge set is $\{e_{1},e_{2},e_{3},e_{4},e_{5},e_{6}\}$.

Both $e_{1}$ and $e_{6}$ have endpoints $v_{1}$ and $v_{2}$, thus they are multiple edges.

Simple Graphs

A graph without multiple edges and without loops is called a simple graph.

In a simple graph we can name the edges by their endpoints:

In the following graph, the edge set is $\{v_{1}v_{2},v_{1}v_{3},v_{2}v_{4}, v_{3}v_{4}, v_{4}v_{5}\}$

Definition 1.1.8. 

• The complement $\overline{G}$ of a simple graph $G$ is the simple graph with vertex set $V(G)$ defined by $uv\in E(\overline{G})$ if and only if $uv\notin E(G)$.
• A clique in a graph is a set of pairwise adjacent vertices.
• An independent set (or stable set) in a graph is a set of pairwise nonadjacent vertices.

Example. 

$G$

$\overline{G}$

Example 1.1.7. Consider a group of 5 people. Is there always a group of three who are all mutual acquaintances or mutual strangers? In other words, for every simple graph on 5 vertices $G$, does either $G$ or $\overline{G}$ contain a clique of size 3?

Definition 1.1.10. A graph $G$ is bipartite if $V(G)$ is the union of two disjoint (possibly empty) independent sets called partite sets of $G$.

Example. 

Definition 1.1.15.

• A path is a simple graph whose vertices can be ordered so that two vertices are adjacent if and only if  they are consecutive in the list.
• A cycle is a graph with an equal number of vertices and edges whose vertices can be placed around a circle so that two vertices are adjacent if and only if they appear consecutively along a circle.

Definition 1.1.16.

• A subgraph of a graph $G$ is a graph $H$ such that $V(H)\subseteq V(G)$ and $E(H)\subseteq E(G)$ and the assignment of endpoints in $H$ is the same as in $G$. We write $H\subseteq G$ and say "$G$ contains $H$".
• A graph $G$ is connected if each pair of vertices in $G$ belongs to a path contained in $G$.

Example. 

Matrices and IsoMorphism

Consider the simple graph with

$$V(G) = \{a,b,c,d,e\}\quad\text{and}\quad E(G) = \{ab,bc,cd,ac,ae\}$$

Definition 1.1.17. Let $G$ be a loopless graph with vertex set $V(G) = \{v_{1},\ldots,v_{n}\}$ and edge set $E(G) = \{e_{1},\ldots,e_{m}\}$. The adjacency matrix of $G$, written $A(G)$ is the $n\times n$ matrix with entry in row $i$, column $j$ is the number of edges in $G$ with endpoints $\{v_{i},v_{j}\}$. The incidence matrix $M(G)$ is the $n\times m$ matrix in which the entry in row $i$, column $j$ is $1$ if $v_{i}$ is an endpoint of edge $e_{j}$, and zero otherwise.

If vertex $v$ is an endpoint of edge $e$, then $v$ and $e$ are incident. The degree of a vertex (in a loopless graph) is the number of incident edges.

Example. Consider the simple graph with

$$V(G) = \{a,b,c,d,e\}\quad\text{and}\quad E(G) = \{ab,bc,cd,ac,ae\}$$

$$A(G) = \left[\begin{array}{ccccc} 0 & 1 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0\end{array}\right]$$

$$M(G) = \left[\begin{array}{ccccc} 1 & 0 & 0 & 1 & 1\\ 1 & 1 & 0 & 0 & 0\\ 0 & 1 & 1 & 1 & 0\\ 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 1\end{array}\right]$$

$$A(G_{1}) = \left[ \begin{array}{cccccc} 0 & 1 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 1 & 1 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \end{array} \right]$$

$$A(G_{2}) = \left[ \begin{array}{cccccc} 0 & 1 & 1 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \end{array} \right]$$

Are these graphs "the same"?

Graph isomorphism

Definition 1.1.20. An isomorphism from a simple graph $G$ to a simple graph $H$ is a bijection $f:V(G)\to V(H)$ such that $uv\in E(G)$ if and only if $f(u)f(v)\in E(H)$. We say that "$G$ is isomorphic to $H$", written $G\cong H$, if there is an isomorphism from $G$ to $H$.

Consider the simple graph $G$ with $$V(G) = \{a,b,c,d,e\}$$ and $$E(G) = \{ab,bc,cd,ac,ae\}$$

Consider the simple graph $H$ with $$V(H) = \{a,b,c,d,e\}$$ and $$E(H) = \{bc,cd,de,bd,ba\}.$$

Proposition. If $f$ is an isomorphism from $G$ and $H$, then for any $v\in V(G)$ the degrees of $v$ and $f(v)$ are the same. Moreover, if

$$d_{1},d_{2},\ldots,d_{n}\quad\text{and}\quad d_{1}',d_{2}',\ldots,d_{n}'$$

are the lists of the degrees of their vertices of $G$ and $H$, respectively, listed in nonincreasing order, then these lists are identical.

Proof. Consider a vertex $v\in V(G)$ with degree $d$. This means that there are $d$ other vertices $v_{1},v_{2},\ldots, v_{d}$ that are adjacent to $v$, that is $vv_{i}\in E(G)$ for each $i\in\{1,2,\ldots,d\}$. By the definition of an isomorphism, the vertices adjacent to $f(v)$ are exactly $f(v_{1}),\ldots,f(v_{d})$. Thus $f(v)$ has degree $d$. $\Box$

Example. None of the following graphs are isomorphic:

Definition. A permutation matrix is a square matrix with entries in $\{0,1\}$ and exactly one $1$ in each row and each column.

Example. Some permutation matrices:

$$\begin{bmatrix} 1 & 0\\ 0 & 1\end{bmatrix}$$

$$\begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\end{bmatrix}$$

$$\begin{bmatrix} 0 & 1 & 0\\ 1 & 0 & 0\\ 0 & 0 & 1\end{bmatrix}$$

$$\begin{bmatrix} 0 & 1 & 0 & 0\\ 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 &  1 & 0\end{bmatrix}$$

Multiplying a matrix by a permutation on the left permutes the rows:

$$\begin{bmatrix} 0 & 1 & 0\\ 0 & 0 & 1\\ 1 & 0 & 0\end{bmatrix}\begin{bmatrix} a & b & c\\ d & e & f\\ g & h & i\end{bmatrix} = \begin{bmatrix}  d & e & f\\ g & h & i\\ a & b & c\end{bmatrix}$$

Multiplying a matrix by a permutation on the right permutes the columns:

$$\begin{bmatrix} a & b & c\\ d & e & f\\ g & h & i\end{bmatrix}\begin{bmatrix} 0 & 1 & 0\\ 0 & 0 & 1\\ 1 & 0 & 0\end{bmatrix} = \begin{bmatrix}  c & a & b\\  f & d & e\\ i & g & h\end{bmatrix}$$

Permutation matrices

Rows: $(1,2,3)\mapsto (3,1,2)$

Columns: $(1,2,3)\mapsto (2,3,1)$

$$\begin{bmatrix} 0 & 0 & 1\\ 1 & 0 & 0\\ 0 & 1 & 0\end{bmatrix}\begin{bmatrix} a & b & c\\ d & e & f\\ g & h & i\end{bmatrix} \begin{bmatrix} 0 & 1 & 0\\ 0 & 0 & 1\\ 1 & 0 & 0\end{bmatrix}= \begin{bmatrix} i & g & h\\ c & a & b\\  f & d & e\end{bmatrix}$$

$1$

$2$

$3$

$1$

$2$

$3$

If we multiply by $P$ on the right and $P^{\top}$ on the left, then we permute the rows and columns in the same way:

Isomorphism: $(1,2,3)\mapsto (2,3,1)$

$$\begin{bmatrix} 0 & 0 & 1\\ 1 & 0 & 0\\ 0 & 1 & 0\end{bmatrix}\begin{bmatrix} 0 & 1 & 1\\ 1 & 0 & 0\\ 1 & 0 & 0\end{bmatrix} \begin{bmatrix} 0 & 1 & 0\\ 0 & 0 & 1\\ 1 & 0 & 0\end{bmatrix}= \begin{bmatrix} 0 & 1 & 0\\ 1 & 0 & 1\\ 0 & 1 & 0\end{bmatrix}$$

Consider permuting the rows and columns of an adjacency matrix:

Theorem. Two simple graphs $G$ and $H$ are isomorphic if and only if there is a permutation matrix $P$ such that

$$A(H) = P^{\top}A(G)P.$$

Thus, if we can find a permutation matrix such that $A(H) = P^{\top}A(G)P$, we can conclude $G$ and $H$ are isomorphic. If $G$ and $H$ are not isomorphic, it is harder to prove.

Definition 1.1.25. An isomorphism class of graphs is an equivalence class of graphs under the isomorphism relation.

Example. Every simple graph on $2$ vertices is isomorphic to one of the following:

So, there are two isomorphism classes of graphs on $2$ vertices.

Example. How many isomorphism classes are there on $3$ vertices?

Connection

in

Graphs

Definition 1.2.2.

• A walk is a list $v_{0},e_{1},v_{1},\ldots,e_{k},v_{k}$ of vertices and edges such that, for      $i\in\{1,2,\ldots,k\}$, the edge $e_{i}$ has endpoints $v_{i-1}$ and $v_{i}$.
• A trail is a walk with no repeated edge.
• A $u,v$-walk or $u,v$-trail has first vertex $u$ and last vertex $v$; the vertices $u$ and $v$ are called its endpoints.
• A $u,v$-path is a path whose endpoints are $u$ and $v$; the other vertices are called internal vertices.
• The length of a walk, trail, path, or cycle is its number of edges. A walk or trail is closed if its endpoints are the same.

Lemma 1.2.5. Every $u,v$-walk contains a $u,v$-path.

Proof. (On the board)

Example. A walk with a repeated vertex:

Definition 1.2.6. A graph is connected if for every pair of vertices $u$ and $v$, there is a $u,v$-path contained in the graph. Otherwise, we say that the graph is disconnected.

If $G$ has a $u,v$-path, then we say that $u$ is connected to $v$ in $G$. 

Definition 1.2.8. The components of a graph $G$ are its maximal connected subgraphs. A component is trivial if it has no edges otherwise it is nontrivial. An isolated vertex is a vertex of degree $0$.

Proposition 1.2.11. Every graph with $n$ vertices and $k$ edges has at least $n-k$ components.

Note, if a graph $G$ contains a $u,v$-path and a $v,w$-path, it is not clear that $G$ contains a $u,w$-path. However, if we concatenate these paths, then we get a $u,w$-walk which contains a $u,w$-path by Lemma 1.2.5. Thus, the connection relation on vertices is transitive. It is also reflexive (vertices are connected to themselves), and symmetric (since paths are reversible).

Proof. Adding an edge to a graph decreases the number of connected components by at most $1$. $\Box$

Definition 1.2.12.

• A cut-edge or cut-vertex of a graph is an edge or vertex, respectively, whose deletion increases the number of components.
• We write $G-e$ or $G-M$ for the subgraph of $G$ obtained by deleting an edge $e$ or a set of edges $M$.
• We write $G-v$ or $G-S$ for the subgraph obtained by deleting a vertex $v$ or a set of vertices $S$.
• An induced subgraph is a subgraph obtained by deleting a set of vertices. 
• We write $G[T]$ for $G-\overline{T}$, where $\overline{T} = V(G)-T$; this is called the subgraph of $G$ induced by $T$.

Theorem 1.2.14. An edge is a cut-edge if and only if it belongs to no cycle.

Proof. (On the board)

Note that this theorem is the statemet that the following two implications about an edge $e$ in a graph $G$ both hold:

$e$ is a cut-edge $\Rightarrow$ $e$ belongs to no cycle

and

$e$ belongs to no cycle $\Rightarrow$ $e$ is a cut-edge

Given an implication $A\Rightarrow B$, it is often useful to consider the equivalent statement $\sim B\Rightarrow \sim A$, which is called the contrapositive. (Here $\sim A$ is the statement "$A$ does not hold".)

Thus, we could instead prove the contrapositives of the above implications

$e$ belongs to a cycle $\Rightarrow$ $e$ is not a cut-edge

and

$e$ is a not a cut-edge $\Rightarrow$ $e$ belongs to a cycle

Eulerian Circuits

Seven Bridges of Konigsberg

Is there a path that crosses all 7 bridges exactly once?

Seven Bridges of Konigsberg

Is there a path that crosses all 7 bridges exactly once?

Seven Bridges of Konigsberg

Is there a path that crosses all 7 bridges exactly once?

Consider a path $P$ in a graph $G$. If there is another path $Q\neq P$ such that $$P\subseteq Q\subseteq G,$$ then we call $Q$ an extension of $P$. Since $G$ only has finitely many vertices, we cannot extend a path forever. If a path $P\subseteq G$ has no extension, then we call $P$ a maximal path.

Proposition 1.2.27. Every even graph decomposes into cycles.

Note: This "decomposition" is a partition of the edges. The cycles may share vertices. Indeed, every vertex of degree $\geq 2$ must be in more than one cycle.

Vertex Degrees  and counting

Definition 1.3.1.

• The degree  of a vertex $v$ in a graph $G$, written $d_{G}(v)$ or $d(v)$, is the number of edges incident to $v$, except that each loop at $v$ counts twice.
• The maximum degree is denoted $\Delta(G)$, the minimum degree is denoted $\delta(G)$, and $G$ is regular if $\Delta(G) = \delta(G)$.
• The graph $G$ is $k$-regular if $\Delta(G) = \delta(G) = k$.
• The neighborhood of $v$, denoted $N_{G}(v)$ or $N(v)$, is the set of vertices adjacent to $v$.

Regular graphs and Degree

$1$

$1$

$2$

$2$

$2$

$4$

$\Delta(G) = 4$ and $\delta(G) = 1$

Example.

The Petersen Graph

Definition 1.1.36. The Petersen graph is the simple graph with vertices consisting of all $2$-element subsets of $\{1,2,3,4,5\}$, that is,  

$$V(G) = \big\{\{1,2\},\{1,3\},\{1,4\},\{1,5\}, \{2,3\}, \{2,4\}, \{2,5\}, \{3,4\},\{3,5\},\{4,5\}\big\},$$

and two vertices are adjacent if the sets are disjoint.

Note that

$\Delta(G) = \delta(G) = 3$

Thus, the Petersen graph is $3$-regular.

Definition 1.1.27.

• The path and cycle with $n$ vertices are denoted $P_{n}$ and $C_{n}$, respectively. An $n$-cycle is a cycle with $n$ vertices.
• A complete graph is a simple graph whose vertices are pairwise adjacent. The complete graph on $n$ vertices is denoted $K_{n}$
• A complete bipartite graph or biclique is a simple bipartite graph such that two vertices are adjacent if and only if they are in different partite sets. When the partite sets have sizes $r$ and $s$, the graph is denoted $K_{r,s}$

Some important graphs

Definition 1.3.2.

• The order of a graph $G$, written $n(G)$, is the number of vertices in $G$. 
• An $n$-vertex graph is a graph of order $n$.
• The size of a graph $G$, written $e(G)$, is the number of edges in $G$. 
• For $n\in\mathbb{N}$, we define $[n]: = \{1,2,3,\ldots,n\}$.

Proposition 1.3.3. (Degree-Sum Formula) If $G$ is a graph, then

$$\sum_{v\in V(G)}d(v) = 2e(G).$$

Proposition 1.3.9. If $k>0$, then a  $k$-regular bipartite graph has the same number of vertices in each partite set.

regular bipartite graphs

Example. Consider the $2$-regular bipartite graph below

6-cycles in the petersen graph

6-cycles in the petersen graph

$\#$ cycles $\geq \#$ vertices:

6-cycles in the petersen graph

$\#$ cycles $\leq \#$ vertices:

Proposition 1.3.11 For a simple graph $G$ with vertices $v_{1},\ldots,v_{n}$ and $n\geq 3$,