# numTriangles -- returns the number of triangles in a graph

## Synopsis

• Usage:
d = numTriangles G
• Inputs:
• G, ,
• Outputs:
• d, an integer, the number of triangles contained in G

## Description

This method is based on work of Francisco-Ha-Van Tuyl, looking at the associated primes of the square of the Alexander dual of the edge ideal. The algorithm counts the number of these associated primes of height 3.

See C.A. Francisco, H.T. Ha, A. Van Tuyl, "Algebraic methods for detecting odd holes in a graph." (2008) Preprint. arXiv:0806.1159v1.

 i1 : R = QQ[x_1..x_6]; i2 : G = graph({x_1*x_2,x_2*x_3,x_3*x_4,x_4*x_5,x_1*x_5,x_1*x_6,x_5*x_6}) --5-cycle and a triangle o2 = Graph{edges => {{x , x }, {x , x }, {x , x }, {x , x }, {x , x }, {x , x }, {x , x }}} 1 2 2 3 3 4 1 5 4 5 1 6 5 6 ring => R vertices => {x , x , x , x , x , x } 1 2 3 4 5 6 o2 : Graph i3 : numTriangles G o3 = 1 i4 : H = completeGraph R; i5 : numTriangles H == binomial(6,3) o5 = true