KU Math 725: Lecture 1/23/2026
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Important classes of graphs
[example=%counter%] Some important families of graphs to keep in mind:
- Paths $P_n$
- $n$ vertices and $n - 1$ edges.
- Cycles $C_n$
- $n$ vertices and $n$ edges.
- Complete graph $K_n$
- $n$ vertices and ${n\choose 2}$ edges.
- Complete bipartite graph $K_{p, q}$.
- $p + q$ vertices and $pq$ edges.
- Empty graph $\overline{K_n}$.
- $n$ vertices and no edges. [/example]
[definition=%counter%] A copy of $\overline{K_n}$ appearing as an induced subgraph of $G$ arises from a set of $n$ vertices for which no pair are adjacent. We call such a subgraph a co-clique or independent set. [/definition]
[example=%counter%] The red vertices in the following graph is a co-clique:
[/example]
Graph operations
[definition=%counter%] Let $G$ be a simple graph. Its complement $\overline{G}$ is the graph obtained by toggling adjacency and non-adjacency. [/definition]
[definition=%counter%] The underlying simple graph $G^S$ of any graph $G$ is obtained by deleting all loops and all but one of each parallel class of edges. [/definition]
[definition=%counter%] The disjoint union $G + H$ is a new graph with vertices $V(G) \sqcup V(H)$ and edge set $E(G) \sqcup E(H)$.
[/definition]
[example=%counter%] $\overline{K_m} + \overline{K_n} = \overline{K_{m+n}}$. More intuitively, $m$ dots on a page and $n$ dots on a page drawn side-by-side is the same thing as $m+n$ dots on a page. [/example]
[example=%counter%] $K_m + K_n = \overline{K_{m,n}}$. [/example]
[definition=%counter%] The join $G * H$ has vertex set $V(G) \sqcup V(H)$ but the edge set is $E(G) \sqcup E(H)$ along with every possible edge between a vertex of $G$ and a vertex of $H$. [/definition]
[example=%counter%] $K_2 * K_3 = K_5$. [/example]
[definition=%counter%] $G - v$ is the graph $G[V(G) - \{v\}]$. [/definition]
[example=%counter%]
[/example]
Handshaking Theorem and Consequences, $d$-regularity
[definition=%counter%] The number of vertices of $G$ is written $n(G)$ or $|G|$ and is the order of $G$.
The number of edges of $G$ is written $e(G)$ or $\|G\|$ and is the size of $G$. [/definition]
[definition=%counter%] The degree of $v \in V(G)$, written $d(v)$ or $d_G(v)$, is the number of edges with $v$ as an endpoint (loops count twice). The minimum and maximum degrees of any vertex in $G$ are written $\delta(G)$ and $\Delta(G)$, respectively. [/definition]
[proposition=%counter% (Handshaking Theorem; 1.6 in Reuven’s notes)] For every graph $G$,
\[\sum_{v\in V(G)} d(v) = 2e(G).\][/proposition]
[proof] The LHS counts each edge twice so we obtain the RHS. [/proof]
[corollary=%counter% (1.7 in Reuven’s notes)] Every graph has an even number of vertices of odd degree. [/corollary]
[proof] Let $G$ be a graph. By the Handshaking Theorem,
\[\sum_{v \in V(G)} d(v) = 2e(G).\]Since $2e(G)$ is even, it follows that there must be an even number of vertices of odd degree otherwise the LHS would be an odd number. [/proof]
[corollary=%counter% (1.8 in Reuven’s notes)] For every vertex $v$, $\delta(G) \le d(v) \le \Delta(G)$, so
\[\delta(G) \le \frac{2e}{n} \le \Delta(G)\]where $n$ is the number of vertices of $G$. [/corollary]
[proof] The fact that $\delta(G) \le d(v) \le \Delta(G)$ holds for all $v$ is just by definition. For the second claim, we sum over all $v \in V(G)$ and apply the Handshaking Theorem to obtain
\[n\delta(G) \le 2e \le n\Delta(G).\]Now just divide through by $n$. [/proof]
[definition=%counter%] A graph is $d$-regular if every vertex is degree $d$. [/definition]
[example=%counter%] $C_n$ is $2$-regular and $K_n$ is $(n-1)$-regular. [/example]
[example=%counter%] An icosohedron has $12$ vertices and is $5$-regular. [/example]
[example=%counter%] The $n$-dimensional cube or hypercube $Q_n$ is defined as follows:
- The vertices of $Q_n$ are bitstrings of length $n$.
- The edges: two vertices are adjacent if they agree in $n - 1$ places.
Equivalently, let $V = 2^[n]$ and $E = \{ST \mid |S \Delta T| = 1\}$.
$Q_m$ is $m$-regular with $2^m$ vertices and $e = m2^{m-1}$. [/example]
[proof=Proof of $e = m2^{m-1}$] By the Handshaking Theorem, $2e/2^m = m$ where we also use the fact that in an $m$-regular graph $\delta = \Delta$. Solving for $e$ gives the desired claim. [/proof]
[remark=%counter%] Equality holds in Corollary 1.8 for regular graphs. [/remark]
[proof] If $G$ is regular, then $\delta(G) = \Delta(G)$. [/proof]
[corollary=%counter% (1.10 in Reuven’s notes)] There are no regular graphs of odd degree and odd order. [/corollary]
[proof] If there was, then there would be an odd number of vertices with odd degree and that would contradict Corollary 1.7. [/proof]