The 
-book graph is defined as the graph
Cartesian product 
,
where 
is a star
graph, 
is the path graph on two nodes,
and 
denotes the graph
Cartesian product. The generalization of the book graph to 
"stacked" pages may be called 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
].
The book graph 
is the simplex graph of the star
graph 
.
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)
|
 |  | ![(x-1)^(n-2)(x+1)^(n-2)[n^2x^2+(x^2-1)^3+n(-1+2x^2-2x^4)]](/images/equations/BookGraph/Inline29.svg) |
(3)
|
 |  | ![([1+3x(x+1)]^n(y-x)+x(y+1){1+x[3+x(3+y)]}^n)/y.](/images/equations/BookGraph/Inline32.svg) |
(4)
|
The corresponding recurrence relations are
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
 |  |  |
(8)
|
The triangular book graph is a different construction consisting of triangles sharing an edge. It is the book graph used in the Ramsey number literature (Rousseau and Sheehan 1978).
