Magic graph
graph whose edges can be labelled by positive integers so that the sum of the labels at any vertex is the same
A magic graph is a graph whose edges are labelled by the first q positive integers, where q is the number of edges, so that the sum over the edges incident with any vertex is the same, independent of the choice of vertex; or it is a graph that has such a labelling. The name "magic" sometimes means that the integers are any positive integers; then the graph and the labelling using the first q positive integers are called supermagic.
A graph is vertex-magic if its vertices can be labelled so that the sum on any edge is the same. It is total magic if its edges and vertices can be labelled so that the vertex label plus the sum of labels on edges incident with that vertex is a constant.
There are a great many variations on the concept of magic labelling of a graph. There is much variation in terminology as well. The definitions here are perhaps the most common.
Comprehensive references for magic labellings and magic graphs are Gallian (1998), Wallis (2001), and Marr and Wallis (2013).
Magic squares
A semimagic square is an n × n square with the numbers 1 to n2 in its cells, in which the sum of each row and column is the same. A semimagic square is equivalent to a magic labelling of the complete bipartite graph Kn,n.
“Magic graph” enters the record as graph whose edges can be labelled by positive integers so that the sum of the labels at any vertex is the same. Crown Archives preserves that source wording while asking what Magic, graph and whose can confirm, complicate or overturn.
Why this record matters
“Magic graph” is worth following because a concise public description often conceals a longer documentary argument. Here, Magic, graph and whose provides the most credible route into that argument.
Named sources, stable identifiers and responsible institutions provide the strongest route from overview to verifiable evidence. The source revision retrieved here is dated Jan 2, 2026. The linked authority identifier is Q17125113. None of the 0 selected statements returned an explicit reference. The first chronological checks are 1998, 2001 and 2013.
A concise general-reference account can conceal disagreements about scope, terminology or the weight assigned to individual sources. The lead is largely declarative, so disagreement and counter-evidence require a deliberate search beyond the opening account. Authority statements aid reconciliation but still require their own references, qualifiers and ranks to be checked.
How to read it
Use the entry as an orientation point, then follow its citations and revision history. Names, dates and institutional relationships should be checked against the original record.
- Subject orientation
- Search vocabulary
- Locating named sources
The closest primary source, responsible institution and strongest cited specialist reference.
Three-step research path
- Establish the record: confirm the title “Magic graph”, its source revision and the description used here.
- Expand the search: follow Magic graph primary sources, Magic graph archive and Magic research across catalogues and specialist indexes.
- Test the account: compare the strongest cited source with the responsible institution’s current record and note any disagreement.
Questions for further research
- Which source most directly establishes the central claim about “Magic graph”?
- What terminology or title could unlock a more precise catalogue search?
- Which institution is responsible for the underlying evidence?
Search terms from this dossier
This entry incorporates text from “Magic graph” on English Wikipedia. Contributors are listed in the page history. Text is available under the Creative Commons Attribution-ShareAlike 4.0 License. Selected authority identifiers and statements are retrieved from Wikidata under CC0; their references and qualifiers remain part of the verification path.