David C. Fisher

Lower bounds on the number of triangles in a graph

For a simple graph G, let f(z) ≡ 1 − c1z + c3z2 − c3z3 + … where ck is the number of complete subgraphs on k nodes in G. Let r(G) be the reciprocal of the smallest real root of f(z). Let λ(Ḡ) be the spectral radius of the complement of G. We show r(G) ⩾ λ(Ḡ) + 1. This is used to show that if c21/4 ⩽ c2 ⩽ c21/3, then a lower bound on the number of triangles in G is c3 ⩾ [9c2c1 − 2c21 − 2(c21 − 3c2)3/2]/27. This improves a bound of Bollobas and is asymptotically sharp. Also, this paper shows that (a corollary of a result from Erdos and Hanini) and the average number of triangles in a graph with c1 nodes and c2 edges is E(c3) = 3/4(C22 − 3c2 + 2)c2/(c31 − 5c1 − 4). These are graphically compared to the best known lower bounds.

The number of words of length n in a graph monoid

(1989). The Number of Words of Length n in a Graph Monoid. The American Mathematical Monthly: Vol. 96, No. 7, pp. 610-614.

Your favorite parallel algorithms might not be as fast as you think

A problem that requires I inputs, K outputs and I computations is to be solved on a d-dimensional parallel processing machine (usually d >

The number of triangles in a K 4 -free graph

Abstract We show that a K4-free graph with e edges has at most (e⧸3) 3 2 triangles. This supercedes a bound of Moon and Moser and is strict when e = 3n2 for any whole number n.

Strict bounds for the period of periodic orbits of difference equations

Abstract Let f be Lipschitz with constant L in a normed space. For k > 0 and given x0, let xk + 1 = xk + f(xk). Suppose x1 ≠ x0, but xn = x0 for some n ⩾ 2. We show that L ⩾ n λ n where λn is the largest eigenvalue of a computable (n − 1) × (n − 1) integer matrix. An example is given with L = n λ n , so the bound is strict. We also find an explicit formula for λn when n is a prime and compare our results to the best bounds for inner product spaces.

Spline quadrature formulas

In this paper we develop quadrature formulas for splines with equispaced knots. For special classes of splines, we give simple explicit expressions for the weights in the quadrature formulas in terms of the zeros of the Euler-Frobenius polynomials and show that these weights are positive. The zeros of these polynomials of odd degree up to 15, are given by Nilson [2] and by Schoenberg and Silliman [4] to a high degree of accuracy. The general quadrature formulas can also be used to obtain the cubic natural spline quadrature formula given in Ahlberg, Nilson and Walsh [ 1, pp. 44-471 and the semicardinal odd order natural spline formula of Schoenberg and Silliman [4]. Two of the quadrature formulas that we derive can be stated as follows. Let S be a spline of odd degree m with knots at the integers 0, l,..., n. Suppose that S’*“(O+) = S’*“(C) = 0, for 2 < 2i < m 1, where ,S”’ denotes the kth derivative of S. Then