More on block intersection polynomials and new applications to graphs and block designs

Abstract

The concept of intersection numbers of order r for t-designs is generalized to graphs and to block designs which are not necessarily t-designs. These intersection numbers satisfy certain integer linear equations involving binomial coefficients, and information on the non-negative integer solutions to these equations can be obtained using the block intersection polynomials introduced by P.J. Cameron and the present author. The theory of block intersection polynomials is extended, and new applications of these polynomials to the studies of graphs and block designs are obtained. In particular, we obtain a new method of bounding the size of a clique in an edge-regular graph with given parameters, which can improve on the Hoffman bound when applicable, and a new method for studying the possibility of a graph with given vertexdegree sequence being an induced subgraph of a strongly regular graph with given parameters.

Upper bounds on the size of transitive subtournaments in digraphs

“…Cameron-Soicher [2] and Soicher [15] proved the following result. See [15] for further properties of block intersection polynomials.…”

On the Clique Number of a Strongly Regular Graph

“…Note that the example in [17] with parameters (65, 32, 15,16) is an example of a (potential) graph satisfying the hypothesis of Theorem 2.1.…”

Designs, Groups and Computing

“…• a subgroup G of S v , to specify that G-orbits of block designs are isomorphism classes (default: G = S v , giving the usual notion of isomorphism); • a subgroup H of G, such that H is required to be a subgroup of the automorphism group of each returned design (default: H = {1}, but specifying a non-trivial H can be a very powerful constraint; see [16,21,32,33]); • whether the user wants a single design with the specified properties (if one exists), a list of G-orbit representatives of all such designs (i.e. isomorphism class representatives as determined by G; this is the default), or a list of distinct such designs containing at least one representative from each G-orbit.…”

“…Each vertex of this graph represents a possible H-orbit of blocks, each with the same specified multiplicity, with two distinct vertices not joined by an edge only when the totality of the blocks they represent cannot be a submultiset of the blocks of a required design. Such a non-edge may be a result of user-specified properties of the required designs, or may be determined by applying block intersection polynomials [12,33]. The graph problem is then handled by the GRAPE [35] function CompleteSubgraphsOfGivenSize, which uses a complicated backtrack search.…”

“…The total time taken for this calculation is about seven minutes on a 3.1 GHz PC running Linux. [ 4,24 ], [ 8,33 ], [ 12,4 ], [ 16,14 ], [ 24,11 ], [ 32,2 ] A further calculation, taking just one second, shows that no design in L is resolvable; equivalently, no semi-Latin square whose dual is in L is the superposition of Latin squares. …”