On groups admitting no integral Cayley graphs besides complete multipartite graphs

Integral complete multipartite graphs

A graph is called integral if all eigenvalues of its adjacency matrix are integers. In this paper, we investigate integral complete r-partite graphsKp1,p2,...,pr =Ka1·p1,a2·p2,...,as ·ps with s=3, 4.We can construct infinite many new classes of such integral graphs by solving some certain Diophantine equations. These results are different from those in the existing literature. For s = 4, we giv...

Integral Quartic Cayley Graphs on Abelian Groups

A graph is called integral, if its adjacency eigenvalues are integers. In this paper we determine integral quartic Cayley graphs on finite abelian groups. As a side result we show that there are exactly 27 connected integral Cayley graphs up to 11 vertices.

Integral Cayley Graphs over Abelian Groups

Let Γ be a finite, additive group, S ⊆ Γ, 0 6∈ S, − S = {−s : s ∈ S} = S. The undirected Cayley graph Cay(Γ, S) has vertex set Γ and edge set {{a, b} : a, b ∈ Γ, a − b ∈ S}. A graph is called integral, if all of its eigenvalues are integers. For an abelian group Γ we show that Cay(Γ, S) is integral, if S belongs to the Boolean algebra B(Γ) generated by the subgroups of Γ. The converse is proven...

On graphs with complete multipartite µ-graphs

Finite groups admitting a connected cubic integral bi-Cayley graph

A graph   is called integral if all eigenvalues of its adjacency matrix  are integers.  Given a subset $S$ of a finite group $G$, the bi-Cayley graph $BCay(G,S)$ is a graph with vertex set $Gtimes{1,2}$ and edge set ${{(x,1),(sx,2)}mid sin S, xin G}$.  In this paper, we classify all finite groups admitting a connected cubic integral bi-Cayley graph.