Many famous graphs are edge-primitive, for example, the Heawood graph, the Tutte–Coxeter graph and the Higman–Sims graph. In this paper we systematically analyse edge-primitive and edge-quasiprimitive graphs via the O’Nan–Scott Theorem to determine the possible edge and vertex actions of such graphs. Many interesting examples are given and we also determine all G-edge-primitive graphs for G an ...

We recall first some known facts on Jones and Kauffman polynomials for links, and on state models for link invariants. We give next an exposition of a recent spin model due to F. Jaeger and which involves the Higman-Sims graph. The associated invariant assigns to an oriented link the evaluation for a = τ 5 and z = 1 of its Kauffman polynomial in the Dubrovnik form, where τ denotes the golden ra...

In two recent papers [Hafner, Elec. J. Combin. 11 (R77) (2004), 1–33; Ilić, Pace and Magliveras, J. Combin. Math. Combin. Comput. 80 (2012), 267–275] it was shown that the Higman-Sims graph Γ can be decomposed into a disjoint union of five double Petersen graphs. In the second of these papers, it was further shown that all such decompositions fall into a single orbit under the action of sporadi...

How dense can every induced subgraph of b nc vertices (0 < 1) of a triangle-free graph of order n be? Tools will be developed to estimate the local density of graphs, based on the spectrum of the graph and on a fractional viewpoint. These tools are used to refute a conjecture of Erd} os et.al. about the local density of triangle-free graphs for a certain range of , by estimating the local densi...

How dense can every induced subgraph of bnc vertices (0 < 1) of a triangle-free graph of order n be? Tools will be developed to estimate the local density of graphs, based on the spectrum of the graph and on a fractional viewpoint. These tools are used to refute a conjecture of Erd} os et.al. about the local density of triangle-free graphs for a certain range of , by estimating the local densit...

We introduce some well known results about permutation groups, strongly regular graphs, design theory and finite geometry. Our goal is the construction of the Higman-Sims group as an index 2 subgroup of the automorphism group of a (100, 22, 0, 6)-srg. To achieve this we introduce some tools from design theory. Some of the arguments here are slightly more general than those given in lectures. Al...