---- > [!definition] Definition. ([[regular graph]]) > A **$k$-regular graph** is a [[graph]] in which each node has [[degree]] $k$. > [!basicexample] > > [!proposition] > A $3$-regular graph must have an even number of nodes. > ![[CleanShot 2023-09-13 at 10.53.44.jpg]] > > > > [!proof] > > Let $G$ be a $3$-regular graph with $n$ nodes and $m$ edges. The sum of [[degree]]s of nodes any [[graph]] is $2m$; in the case of $g$ we know it is also $3n$. Thus $2m=3n$. $2m$ is even and $3$ is odd; $n$ cannot be odd then because the product of odd numbers is odd. So $n$ is even. > ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```