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    \makebox[\textwidth][l]{\textbf{MATH 6404: Applied [Combinatorics and] Graph Theory} \hfill CU Denver} \\
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    \makebox[\textwidth][l]{Spring 2026 \hfill Instructor: Carlos Mart\'inez}
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\newcommand{\lecturenumber}{$2$}
\newcommand{\lecturetitle}{More on Graphs}
%\newcommand{\scribename}{Carlos Mart\'inez}
\newcommand{\scribename}{Mark Johnson}
\newcommand{\lecturedate}{January 26, 2026}

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\section{Degree}

\subsection{Undirected Graphs}
For an undirected graph $G=(V,E)$, for each $u \in V$, $\textrm{deg}(u)$ is the number of edges adjacent to it.
Formally,
\begin{equation*}
    \textrm{deg}(u):=|\{e=\{v, w\}\in E : u \in e\}|.
\end{equation*}
Note the following terminology:
\begin{itemize}
    \item 
    If $\textrm{deg}(u)=1$, $u$ is a leaf.
    \item 
    If $\textrm{deg}(u)=0$, $u$ is isolated.
\end{itemize}

\subsection{Directed Graphs}
For a directed graph, there are two different notions of degree.
The in-degree (out-degree) of $u \in V$ is the number of edges going into (out of) $u$.
Formally,
\begin{equation*}
    \textrm{deg}^{-}(u)\coloneqq |\{e = (v, w)\in E: w=u\}|
\end{equation*}
and
\begin{equation*}
    \textrm{deg}^{+}(u)\coloneqq |\{e = (v, w)\in E: v=u\}|.
\end{equation*}

\subsection{Examples}
Consider the undirected graph in Figure~\ref{fig: undirected}.
\begin{figure}[ht]
    \centering
    \resizebox{0.333\linewidth}{!}{%
        \begin{tikzpicture}
            \Vertex[x=0,y=0,label=$1$]{1}
            \Vertex[x=-2,y=2,label=$2$]{2}
            \Vertex[x=2,y=2,label=$3$]{3}
            \Edge(1)(2)
            \Edge(2)(3)
        \end{tikzpicture}
    }
\caption{An undirected graph.}
\label{fig: undirected}
\end{figure}
Then, we have
\begin{itemize}
    \item $\textrm{deg}(1) = 1$,
    \item $\textrm{deg}(2) = 2$, and
    \item $\textrm{deg}(3) = 1$.
\end{itemize}

Similarly, consider the directed graph in Figure~\ref{fig: directed}.
\begin{figure}[ht]
    \centering
    \resizebox{0.333\linewidth}{!}{%
        \begin{tikzpicture}
            \Vertex[x=0,y=0,label=$1$]{1}
            \Vertex[x=-2,y=2,label=$2$]{2}
            \Vertex[x=2,y=2,label=$3$]{3}
            \Edge[Direct](1)(2)
            \Edge[Direct](2)(3)
        \end{tikzpicture}
    }
\caption{A directed graph.}
\label{fig: directed}
\end{figure}
Then, we have
\begin{itemize}
    \item 
    $\textrm{deg}^{-}(1)=0$ and $\textrm{deg}^{+}(1)=1$,
    \item 
    $\textrm{deg}^{-}(2)=1$ and $\textrm{deg}^{+}(2)=1$, and
    \item 
    $\textrm{deg}^{-}(3)=1$ and $\textrm{deg}^{+}(3)=0$.
\end{itemize}

\section{Subgraphs}
Let $G=(V(G), E(G))$ and $H=(V(H), E(H))$.
$H$ is a subgraph of $G$ if $V(H)\subseteq V(G)$ and $E(H)\subseteq E(G)$.

\section{Union of Graphs}
The union of $G = (V(G), E(G))$ and $H = (V(H), E(H))$ is a graph $W = (V(W), E(W))$ such that
\begin{itemize}
    \item 
    $V(W)=V(G)\cup V(H)$ and
    \item 
    $E(W)=E(G)\cup E(H)$.
\end{itemize}

\section{Paths}

\subsection{Undirected Graphs}
A path is a graph $P=(V(P), E(P))$ where $V(P)$ can be totally ordered such that two nodes are adjacent in $P$ only if they are adjacent in the order.

We can visualize this in Figure~\ref{fig: a path} and Figure~\ref{fig: another path}.
\begin{figure}[ht]
    \centering
    \resizebox{.5\linewidth}{!}{%
        \begin{tikzpicture}
            \Vertex[x=-4,y=0,label=$1$]{1}
            \Vertex[x=-2,y=0,label=$2$]{2}
            \Vertex[x=0,y=0,label=$3$]{3}
            \Vertex[x=2,y=0,label=$4$]{4}
            \Vertex[x=4,y=0,label=$5$]{5}
            \Edge(1)(2)
            \Edge(2)(3)
            \Edge(3)(4)
            \Edge(4)(5)
        \end{tikzpicture}
    }
    \caption{A path.}
    \label{fig: a path}
\end{figure}

\begin{figure}[ht]
    \centering
    \resizebox{.5\linewidth}{!}{%
        \begin{tikzpicture}
            \Vertex[x=-4,y=0,label=$a$]{1}
            \Vertex[x=-2,y=0,label=$d$]{2}
            \Vertex[x=0,y=0,label=$w$]{3}
            \Vertex[x=2,y=0,label=$z$]{4}
            \Vertex[x=4,y=0,label=$f$]{5}
            \Edge(1)(2)
            \Edge(2)(3)
            \Edge(3)(4)
            \Edge(4)(5)
        \end{tikzpicture}
    }
    \caption{Another path.}
    \label{fig: another path}
\end{figure}
In the example, both graphs are paths.
The graph in Figure~\ref{fig: another path} is a a path: to see this algebraically, we can map each node to $\mathbb{N}$ where $a$ maps to $1$, $d$ maps to $2$, $w$ maps to $3$, and so on.
A path can be expressed as:
\begin{itemize}
    \item 
    A sequence of nodes, such as $v_1, v_2, \ldots, v_k$.
    \item 
    A sequence of edges, such as $e_1, e_2, \ldots, e_{k-1}$.
    \item
    A sequence of nodes and edges, such as
    $v_1,e_1,v_2,e_2, \ldots,v_k,e_{k-1}$.
\end{itemize}
Note that $|E(P)| = |V(P)| - 1$.
If $v_1=s$ and $v_k=t$, we say $P$ is an $\{s, t\}-$path.
If $P$ is an $\{s,t\}-$path $P$, then
\begin{itemize}
    \item 
    $\textrm{deg}(s) = 1$,
    \item 
    $\textrm{deg}(t)=1$, and
    \item 
    $\textrm{deg}(u) = 2$ for all $u \in V(P)$ with $u \neq s, t$.
\end{itemize}
    
\subsection{Directed Graphs}
Directed paths have the same formal definition with the additional requirement that
\begin{equation*}
    e_i=(v_i, v_{i+1})
\end{equation*}
for all $i \in [k - 1]$.
In other words, the head of an edge is the same as the tail of the subsequent edge.
Figure~\ref{fig: a directed path} shows a directed path.
\begin{figure}[ht]
    \centering
    \resizebox{.5\linewidth}{!}{%
        \begin{tikzpicture}
            \Vertex[x=-4,y=0,label=$a$]{1}
            \Vertex[x=-2,y=0,label=$d$]{2}
            \Vertex[x=0,y=0,label=$w$]{3}
            \Vertex[x=2,y=0,label=$z$]{4}
            \Vertex[x=4,y=0,label=$f$]{5}
            \Edge[Direct](1)(2)
            \Edge[Direct](2)(3)
            \Edge[Direct](3)(4)
            \Edge[Direct](4)(5)
        \end{tikzpicture}
    }
    \caption{A directed path.}
    \label{fig: a directed path}
\end{figure}

Note that again $|E(P)| = |V(P)| - 1$.
If $P$ is an $(s,t)$-path, then
\begin{itemize}
    \item 
    $\textrm{deg}^{-}(s)=0$ and $\textrm{deg}^{+}(s)=1$, 
    \item 
    $\textrm{deg}^{-}(t)=1$ and $\textrm{deg}^{+}(t)=0$, and
    \item 
    $\textrm{deg}^{-}(u) = \textrm{deg}^{+}(u)= 1$ for all $u \in V(P)$ with $u \neq s, t$.
\end{itemize}

\section{Cycles}
\subsection{Undirected Graphs}
A cycle is a graph $C=(V(C),E(C))$ such that its nodes can be placed around a circle on the plane with two nodes
are adjacent on the circle if and only if they are adjacent in $C$. 
Figure~\ref{fig: a cycle} shows a cycle.
\begin{figure}[ht]
    \centering
    \resizebox{.4\linewidth}{!}{%
        \begin{tikzpicture}
            \Vertex[x=-2,y=0,label=$a$]{1}
            \Vertex[x=2,y=0,label=$d$]{2}
            \Vertex[x=2,y=-2,label=$w$]{3}
            \Vertex[x=0,y=-4,label=$z$]{4}
            \Vertex[x=-2,y=-2,label=$f$]{5}
            \Edge(1)(2)
            \Edge(2)(3)
            \Edge(3)(4)
            \Edge(4)(5)
            \Edge(5)(1)
            
        \end{tikzpicture}
    }
    \caption{A cycle.}
    \label{fig: a cycle}
\end{figure}

Alternatively, a cycle $C$ is a path for which we connect its endpoints with an edge.
Note that $|E(C)| = |V(C)|$ and $\textrm{deg}(u) = 2$ for all $u \in V(C)$.

\subsection{Directed Graphs}
Directed cycles have the same formal definition with the additional requirement of respecting directionality.

\section{Connectivity}
\subsection{Undirected Graphs}

Let $G=(V,E)$ be an undirected graph. 
Two (unordered) nodes $u, v\in V$ are connected if $G$ contains a $\{u, v\}$-path. 
The graph $G$ is connected if all (unordered) pairs $u, v \in V$ are connected.
Figure~\ref{fig: a connected graph} shows a connected graph with multiple $\{u,v\}$-paths.
\begin{figure}[ht]
    \centering
    \resizebox{.4\linewidth}{!}{%
        \begin{tikzpicture}
            \Vertex[x=0,y=0]{1}
            \Vertex[x=2,y=-2,label=$v$]{2}
            \Vertex[x=1,y=-4]{3}
            \Vertex[x=-1,y=-4]{4}
            \Vertex[x=-2,y=-2,label=$u$]{5}
            \Edge(1)(2)
            \Edge(2)(3)
            \Edge(3)(4)
            \Edge(4)(5)
            \Edge(5)(1)
            \Edge(4)(1)
            \draw[red, very thick] (2) -- (3);
            \draw[red, very thick] (3) -- (4);
            \draw[red, very thick] (4) -- (5);
        \end{tikzpicture}
    }
    \caption{A  connected graph.}
    \label{fig: a connected graph}
\end{figure}

\subsection{Directed Graphs}
For a directed graph $G = (V,E)$, there are two notions of connectivity:
\begin{itemize}
    \item 
    $G$ is weakly connected if its corresponding undirected graph is connected.
    For example, the graph in Figure~\ref{fig: a directed path} is weakly connected.
    \item
    $G$ is strongly connected if it contains a $(u,v)$-path for all ordered pairs $u, v \in V$.
\end{itemize}

\section{Connected Components}
\subsection{Undirected Graphs}

A connected component of a graph $G = (V, E)$ is an inclusion-wise maximal connected subgraph of $G$.
Here, inclusion-wise maximal means there does not exist another other subgraph with the same property (i.e., connectivity) that strictly contains it.
For example, Figure~\ref{fig: a graph with three connected components} shows a graph with three connected components, of sizes $4$, $3$, and $1$ from left to right.
\begin{figure}[ht]
    \centering
    \tikzset{
        VertexStyle/.style={draw,circle,fill=blue!25, minimum size=7mm, inner sep=0pt},
        EdgeStyle/.style={thick}
    }
    \begin{tikzpicture}[scale=0.8]
        \Vertex[x=0,  y=0]{1}
        \Vertex[x=2,  y=-2]{2}
        \Vertex[x=0,  y=-4]{3}
        \Vertex[x=-2, y=-2]{4}
        \Edge(1)(2)
        \Edge(2)(3)
        \Edge(3)(4)
        \Edge(4)(1)
        \Edge(3)(1)

        \Vertex[x=6,  y=0]{5}
        \Vertex[x=8,  y=-2]{6}
        \Vertex[x=6,  y=-4]{7}
        \Edge(5)(6)
        \Edge(7)(5)

        \Vertex[x=12, y=0]{8}
    \end{tikzpicture}
    \caption{A graph with three connected components.}
    \label{fig: a graph with three connected components}
\end{figure}

\subsection{Directed Graphs}
A weakly connected component of a directed graph $G = (V,E)$ is an inclusion-wise maximal weakly connected subgraph of $G$.
Similarly, a strongly connected component of $G$ is an inclusion-wise maximal strongly connected subgraph of $G$.

\end{document}