Abstract
Let G be a non-trivial finite group, S ? G \ {e} be a set such that if a 2 S, then a-1 ? S and e be the identity element of G. Suppose that Cay(G, S) is the Cayley graph with the vertex set G such that two vertices a and b are adjacent whenever a-1 ? S. An arbitrary graph is called integral whenever all eigenvalues of the adjacency matrix are integers. We say that a group G is Cayley integral simple whenever every connected integral Cayley graph on G is isomorphic to a complete multipartite graph. In this paper we prove that if G is a non-simple group, then G is Cayley integral simple if and only if G ? Zp2 for some prime number p or G ? Z2 x Z2. Moreover, we show that there exist finite non-abelian simple groups which are not Cayley integral simple.
Highlights
A graph is called integral whenever all eigenvalues of the adjacency matrix are integers
In 1974, Harary and Schwenk have first introduced the notion of an integral graph [8]
The characterization of integral graphs seems very difficult so that it is better to concentrate on some special types of graphs
Summary
A graph is called integral whenever all eigenvalues of the adjacency matrix are integers. They a∈A proved that every atom of the Boolean algebra of subgroups of a finite group G is {b ∈ G| a = b } for some a ∈ G, where the minimal non-empty subsets of a Boolean algebra are called atoms and it is well-known that every element of a Boolean algebra is expressible as a union of atoms It follows that every Cayley graph Cay(G, S) over an abelian group G is integral if and only if S is a finite unions of some atoms. It is shown in [4] that the Boolean algebra generated by the subgroups of an arbitrary finite group is contained in the Boolean algebra generated by the integral sets of the group.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
