House of Graphs: A database of interesting graphs

Abstract

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.

Program Fullerene: A software package for constructing and analyzing structures of regular fullerenes

“…The number of regular fullerene isomers N iso (C n ) for a specific number of carbon atoms n increases as . As an illustration of the growth, C 60 has 1812 isomers, and C 180 has 79,538,751 isomers . The order n 9 ‐growth follows from Thurston's parameterization of triangulations on geodesic domes .…”

“…Although the number of isomers is reduced considerably when pentagons are attached to hexagons only—the so‐called isolated pentagon rule (IPR), which increases the thermodynamic stability of fullerenes considerably—the number of IPR isomers still seem to increase asymptotically as n 9 , as shown in Figure . This is likely due to the proportion of IPR to non‐IPR fullerenes growing with increasing n : out of the 1812 C 60 isomers only one is IPR, but out of the isomers for C 180 , 4,071,832—one in twenty—are IPR . As a consequence, for large fullerenes, it becomes difficult to find the thermodynamically most favored isomer.…”

Integer sequence discovery from small graphs

“…Databases originally found only in print, such as the Atlas of Graphs [1], have rapidly expanded to the electronic medium. These databases range from those of mathematical and algorithmic interest[2, 3], to those cataloging structures found in the applied sciences such as ChemSpider[4], RNA topologies[5, 6] or social databases[7, 8, 9]. Due to the rapid growth in the number of unique isomorphic graphs, the currently available databases are specialized in the number of graphs considered; a judicious choice often restricts the study to an interesting and more manageable subset.…”

Secure sets and their expansion in cubic graphs

“…If G is K 4 or K 3 3 then clearly it isˆ -expandable. Hence, let us assume that this is not the case.…”

“…More recently many authors studied the problem of generating cubic graphs, see e.g., [3,5]. The database of cubic graphs of small order can be found in [4]. There are 27 cubic graphs with at most 10 vertices.…”