Finite directed graph

Mathematical statement

Each edge is (u,v) with u≠v; (u,v) and (v,u) are different edges. Edges form a set, so same-direction parallel duplicates are forbidden. In-degrees and out-degrees are separate, each summing over the whole graph to |E|. Successive walk steps must follow orientation. Explicit loop boundaries fail this declared model, not every possible digraph convention.

Representative formula (exact TeX)

D(V,E)

Assumptions

• Vertices are0 through n−1 with n≥1.
• Edges are canonical lexicographically ordered integer pairs, unique and loop-free.

Symbol scope

• D — finite directed graph
• E — ordered edge set

Intuition, definition, and misconceptions

Orientation is part of an edge meaning.

In-degree and out-degree totals each equal the edge count.

Loops and parallel edges need a different declared model.

Do not identify opposite oriented edges or reuse undirected degree conventions.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Binary relations and set projections — An edge is an ordered pair.
• Set — Vertices and edges are sets.
• Cardinality comparison of sets — Degree counts have finite sizes.

Verification boundary

The main slice excludes loops, parallel edges and undirected connectivity conventions.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Add topological ordering and flow networks.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.