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Recipe

Compute a degree sequence from an edge list

Use when: you're given a graph's edge list, possibly with a self-loop, and asked for the degree of some or all of its vertices

Goal: the degree of every vertex asked about, plus which of them are isolated or pendant

Taught in Degree, isolated, and pendant vertices, revisited

Quick recipe

List every edge touching the vertex (or vertices) you need
Tally each vertex's occurrences, counting a self-loop twice
Classify each vertex by its tally

1
Write out the full edge list, and tally directly from it rather than redrawing the graph.

Example

$G$ has vertices $\{A,B,C,D,E\}$ and edges $\{A,B\}, \{A,C\}, \{B,C\}, \{A,D\}$ .

Why: degree is a count of edge-touches, so the edge list is the only data that count actually depends on.
2
For every vertex, count how many times it appears among the listed edges. A self-loop adds $2$ to that one vertex's tally by itself, not $1$ .

Example

$A$ appears in $3$ edges ( $AB, AC, AD$ ), $B$ in $2$ ( $AB, BC$ ), $C$ in $2$ ( $AC, BC$ ), $D$ in $1$ ( $AD$ ), $E$ in $0$ . Separately, a pseudograph vertex $v$ with a single self-loop and nothing else gets $\deg(v) = 2$ , since both ends of that one loop attach to $v$ .
- the edge is an ordinary edge between two distinct vertices — add $1$ to each of the two endpoints
- the edge is a self-loop at one vertex — add $2$ to that vertex alone
Why: both ends of a loop attach to the same vertex, so a loop contributes twice to that one vertex's degree even though only one vertex is involved.
3
Compare each tally to the two named extremes: a tally of $0$ makes that vertex isolated, and a tally of $1$ makes it pendant.

Example

$\deg(D) = 1$ makes $D$ pendant; $\deg(E) = 0$ makes $E$ isolated. $A$ , $B$ , and $C$ sit strictly between the two extremes and get neither name.

Why: isolated and pendant are just names for the two extreme degree values, not a separate count to compute.

Common mistakes

Counting a self-loop once instead of twice.
Forgetting that a self-loop still counts as an edge touching its vertex, so that vertex is not isolated.
Mislabeling a degree- $2$ vertex as pendant by miscounting one of its edges.
Confusing "isolated" (no edges at all) with "disconnected from the rest of the graph" while still having a self-loop or other edge.