Vol 56, No 4 (2017)

We consider undirected graphs without loops and multiple edges. Previously, V. P. Burichenko and A. A. Makhnev [1] found intersection arrays of distance-regular locally cyclic graphs with the number of vertices at most 1000. It is shown that the automorphism group of a graph with intersection array {15, 12, 1; 1, 2, 15}, {35, 32, 1; 1, 2, 35}, {39, 36, 1; 1, 2, 39}, or {42, 39, 1; 1, 3, 42} (such a graph enters the above-mentioned list) acts intransitively on the set of its vertices.

The present paper is one in our series of works on algebraic geometry over arbitrary algebraic structures, which focuses on the concept of geometrical equivalence. This concept signifies that for two geometrically equivalent algebraic structures \( \mathcal{A} \) and ℬ of a language L, the classification problems for algebraic sets over \( \mathcal{A} \) and ℬ are equivalent. We establish a connection between geometrical equivalence and quasiequational equivalence.

We study computable reducibility of computable metrics on R induced by reducibility of their respective Cauchy representations. It is proved that this ordering has a subordering isomorphic to an arbitrary countable tree. Also we introduce a weak version of computable reducibility and construct a countable antichain of computable metrics that are incomparable with respect to it. Informally, copies of the real line equipped with these metrics are pairwise homeomorphic but not computably homeomorphic.

P-stable polygons are studied. It is proved that the property of being (P, s)-, (P, a)-, and (P, e)-stable for the class of all polygons over a monoid S is equivalent to S being a group. We describe the structure of (P, s)-, (P, a)-, and (P, e)-stable polygons _SA over a countable left zero monoid S and, under the condition that the set A \ SA is indiscernible, over a right zero monoid.