Abstract
We give a sharp bound on the number of triangles in a graph with fixed number of edges. We also characterize graphs that achieve the maximum number of triangles. Using the upper bound on number of triangles, we give a sharp bound for the size of the Schur multiplier of special p-group of all ranks. We also improve existing bounds for the size of Schur multiplier of p-groups of class
, and for p-groups of coclass r.
Acknowledgments
The authors asked Marcin Mazur whether he knows a bound for the set where X is a subset of
. Mazur informed us that there is a bijection between
and the triangles in a graph with vertices
and edges X and pointed us to Rivin’s paper [Citation15]. We are grateful to him for this input as it simplified the exposition in Section 3. We also thank Robert F. Morse for providing us with the GAP code and computing the examples mentioned in Remark 2. We thank Komma Patali for helpful discussions. We thank the referee for their suggestions.