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In 1988, Beck (J Algebra 116:208–226, 1988) conjectured that the chromatic number and clique number is the same for the graphs associated to any commutative ring with unity. In 1993, Anderson and Naseer (J Algebra 159:500–514, 1993) disproved the conjecture with a counterexample. In this paper, we find the clique number and bounds for the chromatic number of the graph associated with the finite product of commutative rings with unity. Also we show that some of the results, proved in (Anderson and Naseer in J Algebra 159:500–514, 1993, Beck in J Algebra 116:208–226, 1988) are consequences of our results. In addition, we have constructed a new family of counterexamples of the above conjecture. [ABSTRACT FROM AUTHOR]

We study a class of edge-coloured graphs, including the chamber systems of buildings and other geometries such as affine planes, from which we build homogeneous coherent configurations (also known as association schemes). The condition we require is that the graph be endowed with a certain distance function, taking its values in the adjacency algebra (itself generated by the adjacency operators). When all the edges are of the same colour, the condition is equivalent to the graph being distance-regular, so our result is a generalization of the classical fact that distance-regular graphs give rise to association schemes. The Bose-Mesner algebra of the coherent configuration is then isomorphic to the adjacency algebra of the graph. The latter is more easily computed, and comes with a "small" set of generators, so we are able to produce examples of Bose-Mesner algebras with particularly simple presentations. When a group G acts "strongly transitively", in a certain sense, on a graph, we show that a distance function as above exists canonically; moreover, when the graph is a building, we show that strong transitivity is equivalent to the usual condition of transitivity on pairs of incident chambers and apartments. As an example, we show that the action of the Mathieu group M 24 on its usual geometry is not strongly transitive. We study affine planes in detail. These are not buildings, yet the machinery developed allows us to state and prove some results which are directly analogous to classical facts in the theory of projective planes (which are buildings). In particular, we prove a variant of the Ostrom-Wagner Theorem on Desarguesian planes which holds in both the projective and the affine case. [ABSTRACT FROM AUTHOR]

We define modular Terwilliger algebras of association schemes, Terwilliger algebras over a positive characteristic field, and consider basic properties. We give a condition for the modular Terwilliger algebra to be non-semisimple. We show that the dimension of a Terwilliger algebra of a Johnson scheme depends on the characteristic of the coefficient field. We also give some other examples. [ABSTRACT FROM AUTHOR]

Let Γ be a finite tree. Fix a base vertex x 0 of Γ and let T = T (x 0) be the Terwilliger algebra of Γ with respect to x 0 . Denote by H the group of automorphisms of Γ that fix x 0 , and let S = End H (V) be the centralizer algebra of H, where V = C X is the standard module of T with X the underlying vertex set of Γ . It is obvious that T is contained in S. We show how large the gap is between T and S by comparing irreducible representations of them; in particular we find precisely when T = S holds. [ABSTRACT FROM AUTHOR]