Quartic integral Cayley graphs

Abstract

We give exhaustive lists of connected 4-regular integral Cayley graphs and connected 4-regular integral arc-transitive 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 non-isomorphic choices for Γ and/or different choices for S. A graph is arc-transitive if its automorphism group acts transitively on the ordered pairs of adjacent vertices. Up to isomorphism, there are 27 quartic integral graphs that are arc-transitive. Of these 27 graphs, 16 are bipartite and 16 are Cayley graphs. By taking quotients of our Cayley or arc-transitive 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.

Eigenvalues of Cayley graphs

“…In 2015, Minchenko and Wanless [243] proved that up to isomorphism there are precisely 32 connected 4-regular integral Cayley graphs, 17 of which are bipartite. They also proved that there are exactly 27 arc-transitive 4-regular integral graphs of degree 4, 16 of which are bipartite.…”

On finite groups all of whose cubic Cayley graphs are integral

“…Since G ∈ G 4 , Cay( a, b , {a, a −1 , b, b −1 }) is a quartic connected integral graph. Note that all quartic connected Cayley integral graphs were obtained in [19]. Observe that a b = a −1 .…”

Integral graphs obtained by dual Seidel switching

“…In 2015, M. Minchenko and I. M. Wanless [12] investigated 4-regular integral Cayley graphs. It was shown that up to isomorphism, there are 32 connected 4-regular integral Cayley graphs (17 of them are bipartite) and 27 connected 4-regular integral arc-transitive graphs (16 of them are bipartite, and 16 of them are Cayley graphs).…”