We classify the distance-regular Cayley graphs with least eigenvalue −2 and diameter at most three. Besides sporadic examples, these comprise of the lattice graphs, certain triangular graphs, and line graphs of incidence graphs of certain projective planes. In addition, we classify the possible connection sets for the lattice graphs and obtain some results on the structure of distance-regular Cayley line graphs of incidence graphs of generalized polygons.

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.

Abstract. Let G be a finite group, S ⊆ G \ {1} be a set such that if a ∈ S, then a −1 ∈ S, where 1 denotes the identity element of G. The undirected Cayley graph Cay(G, S) of G over the set S is the graph whose vertex set is G and two vertices a and b are adjacent whenever ab −1 ∈ S. The adjacency spectrum of a graph is the multiset of all eigenvalues of the adjacency matrix of the graph. A graph is called integral whenever all adjacency spectrum elements are integers. Following Klotz and Sander, we call a group G Cayley integral whenever all undirected Cayley graphs over G are integral. Finite abelian Cayley integral groups are classified by Klotz and Sander as finite abelian groups of exponent dividing 4 or 6. Klotz and Sander have proposed the determination of all non-abelian Cayley integral groups. In this paper we complete the classification of finite Cayley integral groups by proving that finite non-abelian Cayley integral groups are the symmetric group S 3 of degree 3, C 3 ⋊ C 4 and Q 8 × C n 2 for some integer n ≥ 0, where Q 8 is the quaternion group of order 8.

Let G be a finite group, and S be a subset of G\{1} such that S = S −1 .Suppose that Cay(G, S) is the Cayley graph on G with respect to the set S which is the graph whose vertex set is G and two vertices a, b ∈ G are adjacent whenever ab −1 ∈ S. The adjacency spectrum Spec(Γ) of a graph Γ is the multiset of eigenvalues of its adjacency matrix. A graph Γ is called "determined by its spectrum" (or for short DS) whenever if a graph Γ ′ has the same spectrum as Γ, then Γ ∼ = Γ ′ . We say that the group G is DS (Cay-DS, respectively) whenever if Γ is a Cayley graph over G and Spec(Γ) = Spec(Γ ′ ) for some graph (Cayley graph,In this paper, we study finite DS groups and finite Cay-DS groups. In particular we prove that all finite DS groups are solvable and all Sylow p-subgroups of a finite DS group is cyclic for all p ≥ 5. We also give several infinite families of non Cay-DS solvable groups. In particular we prove that there exist two cospectral non-isomorphic 6-regular Cayley graphs on the dihedral group of order 2p for any prime p ≥ 13.