6 research outputs found
Generalized Paley graphs equienergetic with their complements
We consider generalized Paley graphs , generalized Paley sum
graphs , and their corresponding complements
and , for . Denote by
either or . We compute the spectra of
and and from them we obtain the spectra of
and also. Then we show that, in the
non-semiprimitive case, the spectrum of and
with prime can be recursively obtained, under certain
arithmetic conditions, from the spectrum of the graphs and
for any , respectively. Using the spectra of
these graphs we give necessary and sufficient conditions on the spectrum of
such that and are
equienergetic for . In a previous work we have classified all bipartite
regular graphs and all strongly regular graphs
which are complementary equienergetic, i.e.\@ and are
equienergetic pairs of graphs. Here we construct infinite pairs of
equienergetic non-isospectral regular graphs which
are neither bipartite nor strongly regular.Comment: 22 page
Multicoloured Random Graphs: Constructions and Symmetry
This is a research monograph on constructions of and group actions on
countable homogeneous graphs, concentrating particularly on the simple random
graph and its edge-coloured variants. We study various aspects of the graphs,
but the emphasis is on understanding those groups that are supported by these
graphs together with links with other structures such as lattices, topologies
and filters, rings and algebras, metric spaces, sets and models, Moufang loops
and monoids. The large amount of background material included serves as an
introduction to the theories that are used to produce the new results. The
large number of references should help in making this a resource for anyone
interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will
appear in physic