 TOPICS  # Book Graph The -book graph is defined as the graph Cartesian product , where is a star graph and is the path graph on two nodes. The generalization of the book graph to "stacked" pages is the -stacked book graph.

Special cases of the -book graph are summarized below. graph 1 square graph 2 domino graph

Precomputed properties of book graphs are implemented in the Wolfram Language as GraphData[ "Book", m ].

Book graphs of the form do not satisfy the parity condition for gracefulness and hence are ungraceful (Gallian 2018). Maheo (1980) proved that is graceful and conjectured that is graceful for all positive integer . Delorme (1980) provided a simpler graceful labeling for together with a graceful labeling for , thus establishing the conjecture.

The book graph has chromatic polynomial, independence polynomial, matching polynomial, and rank polynomial given by   (1)   (2)   (3)   (4)

The corresponding recurrence relations are   (5)   (6)   (7)   (8)

Graph Cartesian Product, Stacked Book Graph, Star Graph

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## References

Delorme, D. "Two Sets of Graceful Graphs." J. Graph Th. 4, 247-250, 1980.Gallian, J. "Dynamic Survey of Graph Labeling." Elec. J. Combin. DS6. Dec. 21, 2018. https://www.combinatorics.org/ojs/index.php/eljc/article/view/DS6.Maheo, M. "Strongly Graceful Graphs." Disc. Math. 29, 39-46, 1980.White, A. T. "Imbedding Problems in Graph Theory." Ch. 6 in Graphs of Groups on Surfaces: Interactions and Models (Ed. A. T. White). Amsterdam, Netherlands: Elsevier, p. 49, 2001.

Book Graph

## Cite this as:

Weisstein, Eric W. "Book Graph." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BookGraph.html