Projects per year
Abstract
We give exhaustive lists of connected 4regular integral Cayley graphs and connected 4regular integral arctransitive graphs. An integral graph is a graph for which all eigenvalues are integers. A Cayley graph Cay(Γ, S) for a given group Γ and connection set S ⊂ Γ is the graph with vertex set Γ and with a connected to b if and only if ba−1 ∈ S. Up to isomorphism, we find that there are 32 connected quartic integral Cayley graphs, 17 of which are bipartite. Many of these can be realized in a number of different ways by using nonisomorphic choices for Γ and/or different choices for S. A graph is arctransitive if its automorphism group acts transitively on the ordered pairs of adjacent vertices. Up to isomorphism, there are 27 quartic integral graphs that are arctransitive. Of these 27 graphs, 16 are bipartite and 16 are Cayley graphs. By taking quotients of our Cayley or arctransitive graphs we also find a number of other quartic integral graphs. Overall, we find 9 new spectra that can be realised by bipartite quartic integral graphs.
Original language  English 

Pages (fromto)  381408 
Number of pages  28 
Journal  Ars Mathematica Contemporanea 
Volume  8 
Issue number  2 
Publication status  Published  2015 
Keywords
 Graph spectrum
 integral graph
 Cayley graph
 arctransitive
 vertextransitive bipartite double cover
 voltage assignment
 graph homomorphism
Projects
 1 Finished

Extremal Problems in Hypergraph Matchings
Wanless, I., Greenhill, C. & Aharoni, R.
Australian Research Council (ARC), University of New South Wales (UNSW)
3/01/12 → 31/12/14
Project: Research