In search of a poset structure to the regular exceptional graphs - Barbedo, Inês (2014)
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. An exceptional graph is a connected graph with least eigenvalue greater than or equal to -2 which is not a generalized line graph, and it is shown that the set of regular exceptional graphs is partitioned in three layers. The idea of a recursive construction of regular exceptional graphs is proposed in [1]. With a new technique we prove that all regular exceptional graphs from the 1st and 2nd layer could be produced by this technique. The new recursive technique is based on the construction of families of regular graphs, where each regular graph is obtained by a (k,t)-extension defined by a k- regular graph H such that V(H) is a (k,t)-regular set of the extended regular graph. The process of extending a graph is reduced to the construction of the incidence matrix of a combinatorial 1-design, and these extensions induce a partial order. Considering several rules to reduce the production of isomorphic graphs, each exceptional regular graph is constructed by a (0,2)-extension. Based on this construction, an algorithm to produce the regular exceptional graphs of the 1st and 2nd layer is introduced and the corresponding poset is presented, using its Hasse diagram.