The construction of the poset of regular execeptional graphs using equitable partitions - Barbedo, Inês (2013)
An exceptional graph is a connected graph with least eigenvalue greater than or equal to -2 which is not a generalized line graph. It is shown that the set of regular exceptional graphs is partitioned in three layers. A (k,t)-regular set is a subset of the vertices of a graph, inducing a k-regular subgraph such that every vertex not in the subset has t neighbors in it. A new recursive construction of regular exceptional graphs is proposed, where each regular exceptional graph of the first and the second layer is constructed by a (0,2)-regular set extension. In this talk we present an algorithm based on this recursive construction and show that this technique induces a partial order relation on the set of regular exceptional graphs. The process of extending a graph is reduced to the construction of the incidence matrix of a combinatorial 1-design, considering several rules to prevent the production of isomorphic graphs, and we show that each regular exceptional graph has an equitable partition which, by this construction technique, is extended with a new element, the set of the additional vertices. The recursive construction is generalized to the construction of arbitrary families of regular graphs, by extending a regular graph G with another regular graph H such that V(H) is a (k,t)-regular set of the regular graph produced. This technique is used to construct the exceptional regular graphs of the third layer. The Hasse diagrams of the posets of the three layers are presented.