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Abstract: This paper is devoted to the conjecture saying that, for any connected locally finite graph and any vertex-transitive group of automorphisms of , at least one of the following assertions holds: (1) There exists an imprimitivity system of on with finite (maybe one-element) blocks such that the stabilizer of a vertex of the factor graph in the induced group of automorphisms is finite. (2) The graph is hyperbolic (i.e., for some positive integer , the graph defined by and contains the regular tree of valency 3). Our approach to the conjecture consists in fixing a finite permutation group and considering the conjecture under the assumption that the stabilizer of a vertex of in induces on the neighborhood of the vertex a group permutation isomorphic to . In the paper we elaborate a method (the modified track method) which allows us to prove the conjecture for many groups . The paper consists of two parts. The present first part of the paper involves results on which the modified track method arguments are based, and a few first applications of the method. The second part is devoted to applications of the modified track method. [Copyright &y& Elsevier]

In this paper Seidel matrices are studied, and their spectrum and several related algebraic properties are determined for order n ≤ 13 . Based on this Seidel matrices with exactly three distinct eigenvalues of order n ≤ 23 are classified. One consequence of the computational results is that the maximum number of equiangular lines in R 12 with common angle 1 ∕ 5 is exactly 20 . [ABSTRACT FROM AUTHOR]

Abstract: Let be a simple graph satisfying the Ore-condition with . In this paper, we prove that is -connected for any abelian group with if and only if is not one of the 12 completely described graphs. [Copyright &y& Elsevier]

Abstract: In this paper, we compute the congruences of Catalan and Motzkin numbers modulo 4 and 8. In particular, we prove the conjecture proposed by Deutsch and Sagan that no Motzkin number is a multiple of 8. [Copyright &y& Elsevier]

Abstract: A language is called a relatively regular language if its syntactic monoid has finite ideals. In this paper, we show that there are close relationships between the relatively regular languages and some other classes of languages such as (generalized) disjunctive languages, fd-domains and 2-codes. In particular, we prove that every relatively regular 2-code is thin. Thus, the well known result of Bestel and Perrin in 1985 becomes an easy corollary of our result. [Copyright &y& Elsevier]

Abstract: In this paper, we consider a relation between the Hall–Janko graph and the Witt system . We will reconstruct the Hall–Janko graph from the group and the hexacode. [Copyright &y& Elsevier]

Abstract: Consider a polar space weakly embedded in a projective space . A secant of is the intersection of the point set of with a line of spanned by two non-collinear points of . The geometry consisting of the points of and as lines the secants is a so-called Delta space. In this paper we give a characterization of this and some related geometries. [Copyright &y& Elsevier]

Abstract: In this paper, we show that the stabilizer subgroup of for a without multiple roots can be derived from the stabilizer of . As an application, we construct a family of 3-designs such as , where is a prime power such that and . [Copyright &y& Elsevier]

Abstract: A graph is called circular super-critical if for every vertex of . In this paper, analogous to a result of Dirac on chromatic critical graphs, a sharp lower bound on the vertex degree of circular super-critical graphs is proved. This lower bound provides a partial answer to a question of X. Zhu [The circular chromatic number of induced subgraphs, J. Combin. Theory Ser. B 92 (2004) 177–181]. Some other structural properties of circular super-critical graphs are also presented. [Copyright &y& Elsevier]

Abstract: Let be a convex polytope in the Euclidean space . Consider the group generated by reflections in the facets of . We say that generates a reflection group . The spherical simplices generating simply-laced reflection groups were described by P.J. Cameron, J.J. Seidel and S.V. Tsaranov [Signed graphs, root lattices and Coxeter groups, J. Algebra 164 (1994) 173–209]. The simplices generating the remaining spherical reflection groups were listed in [A. Felikson, Spherical simplices generating discrete reflection groups, Sb. Math. 195 (3–4) (2004) 585–598, arXiv:math.MG/0212244]. In the present paper, we find all Euclidean simplices generating discrete reflection groups. [Copyright &y& Elsevier]

Abstract: In [B. De Bruyn, P. Vandecasteele, Valuations and hyperplanes of dual polar spaces, J. Combin. Theory Ser. A 112 (2005) 194–211], we introduced the class of the SDPS-valuations of dual polar spaces. We showed that these valuations and all their extensions give rise to hyperplanes of dual polar spaces. We call these hyperplanes SDPS-hyperplanes. In the present paper, we show that a hyperplane of a thick dual polar space is an SDPS-hyperplane if and only if every hex not contained in intersects in either a singular hyperplane or the extension of an ovoid. [Copyright &y& Elsevier]

Abstract: Whereas results about the number of irreducible modular representations of finite groups are known, those of Bose–Mesner algebras of association schemes are not well understood. In this paper, we consider the number of irreducible modular representations of the adjacency algebra of some and polynomial association schemes. These results should be helpful for investigating the structure of association schemes. [Copyright &y& Elsevier]

Abstract: Let be a distance-regular graph of diameter . Suppose that does not have any induced subgraph isomorphic to . In this case, the length of a shortest reduced circuit in is called the geometric girth of . Except for ordinary polygons, all known examples have a property that in general, and if . Is there an absolute constant bound on the geometric girth of a distance-regular graph with valency at least three? This is one of the main problems in the field of distance-regular graphs. P. Terwilliger defined an algebra with respect to a base vertex , which is called a subconstituent algebra or a Terwilliger algebra. The investigation of irreducible -modules and their thin property proved to be a very important tool to study structures of distance-regular graphs. B. Collins proved that if every irreducible -module is thin then is at most 8, and if , then and is a generalized octagon. In this paper, we prove the same result under an assumption that every irreducible -module of endpoint at most 3 is thin. [Copyright &y& Elsevier]

Abstract: In this paper we deal with equitable partitions of association schemes. We try to generalize a result in group theory and show examples that a generalization of a certain property conjectured for permutation groups does not hold for association schemes. [Copyright &y& Elsevier]