In this note we present House of Graphs (http://hog.grinvin.org) which is a new database of graphs. The key principle is to have a searchable database and offer -next to complete lists of some graph classes -also a list of special graphs that already turned out to be interesting and relevant in the study of graph theoretic problems or as counterexamples to conjectures. This list can be extended by users of the database.

We prove the nonexistence of a distance-regular graph with intersection array {74, 54, 15; 1, 9, 60} and of distance-regular graphs with intersection arrays {4r(3) + 8r(2) + 6r + 1, 2r(r + 1)(2r + 1), 2r(2) + 2r + 1; 1, 2r(r + 1), (2r + 1)(2r(2) + 2r + 1)} with r an integer and r >= 1. Both cases serve to illustrate a technique which can help in determining structural properties for distance-regular graphs and association schemes with a sufficient number of vanishing Krein parameters

The Perkel graph is a distance-regular graph of order 57, degree 6 and diameter 3, with intersection array (6, 5, 2; 1, 1, 3). We describe a computer assisted proof that every graph with this intersection array is isomorphic to the Perkel graph. The computer proof relies heavily on the fact that the minimal idempotents for , and their submatrices, are positive semidefinite. To minimize the risk of computer errors we have used two different methods to establish the same theorem and as an added precaution large parts of the corresponding programs were written by different authors. The first method generates plausible subgraphs induced by all vertices at distance 3 from a fixed vertex of and then tries to extend each of the generated graphs to a full graph with the given intersection array. The second method generates possible neighborhoods for a pentagon in . It turns out that every such pentagon can be extended to a Petersen graph in . We then prove mathematically that there is, up to isomorphism, only a single graph with this property.

When the condition of having three equal sides is imposed upon a (convex) spherical quadrangle, the four angles of that quadrangle cannot longer be freely chosen but must satisfy an identity. We derive two simple identities of this kind, one involving ratios of sines, and one involving ratios of tangents, and improve upon an earlier identity by Ueno and Agaoka.The simple form of these identities enable us to further investigate the case in which all of the angles are rational multiples of π and produce a full classification, consisting of 7 infinite classes and 29 sporadic examples. Apart from being interesting in its own right, these quadrangles play an important role in the study of spherical tilings by congruent quadrangles.

A full classification (up to equivalence) of all complete k-arcs in the Desarguesian projective planes of order 27 and 29 was obtained by computer. The resulting numbers of complete arcs are tabulated according to size of the arc and type of the automorphism group, and also according to the type of algebraic curve into which they can be embedded. For the arcs with the larger automorphism groups, explicit descriptions are given.The algorithm used for generating the arcs is an application of isomorphfree backtracking using canonical augmentation, an adaptation of an earlier algorithm by the authors.Part of the computer results can be generalized to other values of q: two families of arcs are presented (of size 12 and size 18) for which the symmetric group S 4 is a group of automorphisms.