KU Math 725: Lecture 1/21/2026

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On this page:

• Basic Definitions
• Graph Isomorphisms and Isomorphism Invariants
• Subgraphs and Induced Subgraphs

Basic Definitions

[definition=%counter%] A graph $G$ is a pair $(V, E)$ where $V$ is a finite nonempty set of vertices and $E$ is a finite set of edges. Each edge $e$ is given by an unordered pair of (possibly equal vertices).

Given an edge $e = (v, w)$, we say:

• $v, w$ are endpoints of $e$;
• $v, w$ are joined by $e$; and
• $e = vw$. [/definition]

[example=%counter%] Let $G$ be the graph:

Here, $V = [5]$ and

[/example]

[definition=%counter%] If $(v, w) \in E(G)$, then we say vertices $v$ and $w$ are adjacent or neighbors. [/definition]

[definition=%counter%] A graph with no loops or multiedges is simple. [/definition]

Graph Isomorphisms and Isomorphism Invariants

[definition=%counter%] Let $G$ and $H$ be two graphs. A graph isomorphism is a bijection $f:V(G) \to V(H)$ such that for every $v, w \in V(G)$

We write $G \cong H$ and say $G$ is isomorphic to $H$ if there exists an isomorphism $G \to H$. [/definition]

[remark=%counter%] Graph isomorphisms $G \to H$ say nothing about how $G$ and $H$ might look on paper. For instance, the following graphs are isomorphic:

[/remark]

[definition=%counter%] An isomorphism invariant is a function $\Psi$ on graphs such that $\Psi(G) = \Psi(H)$ whenever $G \cong H$. Equivalently, a function on equivalence classes of graphs. [/definition]

[example=%counter%] Examples/non-examples of isomorphism invariants:

• The number of vertices is an isomorphism invariant.
• The number of edges is an isomorphism invariant.
• The number of crossings when drawing a graph is NOT an isomorphism invariant (counterexample above).
  • Sidenote: It turns out that the minimum number of crossing is an isomorphism invariant. [/example]

Subgraphs and Induced Subgraphs

[definition=%counter%] Let $G$ be a graph. A subgraph of $G$ is a graph $H$ such that $V(H) \subseteq V(G)$ and $E(H) \subseteq E(G)$. [/definition]

[remark=%counter%] Given $X \subseteq V(G)$ and $F \subseteq E(G)$, $(X, F)$ can fail to be a subgraph of $G$ because not every endpoint of an edge in $F$ may lie in $X$. [/remark]

[definition=%counter%] Let $G$ be a graph and $X \subseteq V(G)$. Then we define $G[X]$ such that $V(G[X]) = X$ and

[/definition]

[example=%counter%] Let $G$ be the graph:

If $X = \{1,3, 4\}$, then the edge set of $G[X]$ consists of $(1, 3), (1, 4)$. [/example]

[remark=%counter%] Not every subgraph is an induced subgraph. As a counterexample, suppose we take $H$ to be such that $V(H) = [4]$ and

in the example above. Then this graph cannot be an induced subgraph of $G$ (such a subgraph would contain the edge $(1, 4)$). [/remark]