Adjacency matrix

Let $Γ$ be a general graph (which may be the underlying graph of a quiver). Given two vertices $𝑣, 𝑤 \in V (Γ)$ , 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 graph

𝑎 𝑖 𝑗 = | Γ (𝑣 𝑖, 𝑣 𝑗) |

develop | 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

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Adjacency matrix

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