Adjacency matrix

Let $Γ$ be a quiver. Given two vertices $𝑣, 𝑤 \in Γ 0$ , the adjacency number $| Γ (𝑣, 𝑤) |$ is the number of arcs $𝑣 \to 𝑤$ .¹ If the vertices are enumerated $𝑣 1, \dots, 𝑣 𝑟$ , then these may be collected in an adjacency matrix $𝐴 Γ = (𝑎 𝑖 𝑗)$ where #m/def/graph

𝑎 𝑖 𝑗 = | Γ (𝑣 𝑖, 𝑣 𝑗) |

#state/develop | #lang/en | #SemBr

Thinking of $A (Γ)$ as a multiset with characteristic function $𝜒 A (Γ)$ , this is just $𝜒 A (Γ) (𝑣, 𝑤)$ . The notation is preferred since it gives a uniform treatment to quivers and their underlying graphs. ↩

Adjacency matrix

Footnotes